ECE161A
Discrete Time Signal Processing
September 25, 2015
Professor T. Javidi
Problem Set #1
Due: Thursday October 1, 2015
I encourage you to work together on your homework but what you hand in must be written up on your own.
1. Consider the system y[n]
Fourier, filtering,
smoothing, and noise
thi
d i
Nuno Vasconcelos
ECE Department, UCSD
p
,
(with thanks to David Forsyth)
Images
the incident light is collected by an image sensor
E
that transforms it into a 2D signal
Po
V
Pi
2
2
2D-DSP
in summary:
imag
Discrete Cosine Transform
Nuno Vasconcelos
UCSD
Discrete Fourier Transform
last classes, we have studied the DFT
due to its computational efficiency the DFT is very
popular
however, it has strong disadvantages for some
applications
it is complex
it h
Least squares and motion
Nuno Vasconcelos
ECE Department, UCSD
Plan for today
today we will discuss motion estimation
this is interesting in two ways
motion is very useful as a cue for recognition, segmentation,
compression, etc.
is a great example of l
Homework Set One
ECE 161
Department of Computer and Electrical Engineering
University of California, San Diego
Nuno Vasconcelos
1. In this problem we consider the perspective projection of lines. A d-dimensional line l is dened by
a point a and a directio
Homework Set Two
ECE 161
Department of Computer and Electrical Engineering
University of California, San Diego
Nuno Vasconcelos
1.
The gure below shows a 2D room illuminated by a single source S at point (0.5, 1) which is not
a point source at innity. We
Least squares
Nuno Vasconcelos UCSD
Model fitting
one common problem in signal processing is to fit a model to a signal
in vision, we typically have a scene it contains some "signal", which is what we are trying to understand about the scene but it also
Filtering, scale, orientation, localization, and texture
Nuno Vasconcelos ECE Department, UCSD (with thanks to David Forsyth)
Beyond edges
we have talked a lot about edges while they are important, it is now recognized that they are not enough for many ob
Edges, interpolation,
templates
t
l t
Nuno Vasconcelos
ECE Department, UCSD
p
,
(with thanks to David Forsyth)
Edge detection
edge detection has many applications
in image p
g processing
g
an edge detector implements the
following steps:
compute gradient
Homework Set Three
ECE 161
Department of Computer and Electrical Engineering
University of California, San Diego
Nuno Vasconcelos
1. Determine whether the following sequences are separable or non-separable. In each case, justify your
answer thoroughly (by
Homework Set Four
ECE 161C
Department of Electrical and Computer Engineering
University of California, San Diego
Nuno Vasconcelos
1. In this question we study an example of why edge detection is such a hard problem. Figure 1
presents an imaging scenario,
Homework Set Five
ECE 161C
Department of Electrical and Computer Engineering
University of California, San Diego
Nuno Vasconcelos
1. Suppose x(n1 , n2 ) is a periodic sequence of period N1 N2 .
a) show that the sequence x(n1 , n2 ) is also periodic with p
Homework Set Six
ECE 161
Department of Computer and Electrical Engineering
University of California, San Diego
Nuno Vasconcelos
1. In class, we saw that an ane transformation is characterized by
x
y
=
a
c
b
d
x
y
+
e
f
.
a) Assume that we computed the mot
Discrete Fourier Transform
Nuno Vasconcelos UCSD
Fourier Transforms
we started by considering the Discrete-Space Fourier Transform (DSFT) the DSFT is the 2D extension of the Discrete-Time Fourier Transform
X (1 , 2 ) = x [n1 , n 2 ]e - j n e - j n
1 1
2
2D DSP
Nuno Vasconcelos
UCSD
Images
the incident light is collected by an image sensor
E
that transforms it into a 2D signal
Po
V
Pi
2
2
2D-DSP
in summary:
image is a N x M array of pixels
each pixel contains three colors
overall, the image is a 2D
Brief Review of Sampling
This material is assumed from ECE101.
Sampling is used, for example, in A/D Conversion, although here we
ignore the effects of quantization.
Discretize a continuous time signal for CDs, computers, etc.
Define the continuous time i
Some Useful Summations
1. Geometric Progression
X
ai =
i=0
2.
N
X
i=0
3.
1
1a
|a| < 1
if a = 0
0
i
N
+
1
if
a=1
a =
1aN+1
otherwise
1a
X
iai =
i=0
a
(1 a)2
|a| < 1
(Take the derivative in Example 1 above)
4.
N
X
i=
i=1
5.
N
X
i=1
i2 =
N(N + 1)
2
N(N + 1
Some Useful Summations
1. Geometric Progression
X
ai =
i=0
2.
N
X
i=0
3.
1
1a
|a| < 1
if a = 0
0
i
N
+
1
if
a=1
a =
1aN+1
otherwise
1a
X
iai =
i=0
a
(1 a)2
|a| < 1
(Take the derivative in Example 1 above)
4.
N
X
i=
i=1
5.
N
X
i=1
i2 =
N(N + 1)
2
N(N + 1
ECE-161C Color
Nuno Vasconcelos ECE Department, UCSD (with thanks to David Forsyth)
Image formation
we are studying the process of image formation two questions
what 3D point projects into pixel (x,y)? what is the light incident on the pixel?
these deter
2D-DSP
Nuno Vasconcelos UCSD
Image formation
we have been studying the process of image formation three questions
what 3D point projects into pixel (x,y)? what is the light incident on the pixel? what is the pixel color?
these determine the image value
ECE-161C
Cameras
Nuno Vasconcelos
ECE Department, UCSD
p
,
Image formation
all image understanding starts with understanding of
image formation:
g
projection of a scene from 3D world into image on 2D plane
2
Image formation
first of all, why do we care a
Radiometry
Nuno Vasconcelos UCSD
Image formation
two components: geometry and radiometry geometry:
pinhole camera point (x,y,z) in 3D scene projected into image pixel of coordinates (x', y ) ( y') according to the perspective projection equation:
x' =
ECE-161C
Nuno Vasconcelos
ECE Department, UCSD
p
,
The course
The course will cover the most important aspects of
image p
g processing and computer vision
g
p
We will cover a lot of ground, at the end of the quarter
you will know how to implement a lot of
Mid-term review
ECE 161C
Electrical and Computer Engineering
University of California San Diego
Nuno Vasconcelos
Spring 2010
1.(20 points) We have seen in class that one popular technique for edge detection is the Canny edge
detector . It contains three p
Edges
Nuno Vasconcelos
ECE Department, UCSD
(with thanks to David Forsyth)
Gradients and edges
for image understanding, one of the problems is that there
is too much information in an image
just smoothing is not good enough
how to detect important (most i
Discrete Fourier Transform
Nuno Vasconcelos UCSD
The Discrete-Space Fourier Transform
as in 1D, an important concept in linear system analysis is that of the Fourier transform the Discrete-Space Fourier Transform is the 2D extension of the Discrete-Time
Radiometry
Nuno Vasconcelos
UCSD
Light
Last class: geometry of image formation
pinhole camera:
point (x,y,z) in 3D scene projected into image pixel of
coordinates (x, y)
according to the perspective projection equation:
x '
=f
y '
x
z
y
z
2
P
Discrete Fourier Transform
Nuno Vasconcelos
UCSD
The Discrete-Space Fourier Transform
as in 1D, an important concept in linear system analysis
is that of the Fourier transform
the Discrete-Space Fourier Transform is the 2D
extension of the Discrete-Time