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 Fou
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 Tra
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 p
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 perspecti
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 Four
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 smoot
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 det
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
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') a
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
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? wh
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 i
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 d
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
=
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) sho
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
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-sepa
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 det
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
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
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 po
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-
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, s
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
Here we want to find v1, the voltage across the 10 k-ohm resistor.
Well use Ohms Law and Kirchhoffs current and voltage laws (KCL and KVL).
First label the all currents and voltages in the circuit.
We
Here we want to find v1, the voltage across the 10 k-ohm resistor.
Well use Ohms Law and Kirchhoffs current and voltage laws (KCL and KVL).
First label the all currents and voltages in the circuit.
We
Here we want to find v0, the voltage across the 10 k-ohm resistor.
Well use Ohms Law and Kirchhoffs current and voltage laws (KCL and KVL).
First label the all currents and voltages in the circuit.
We
Here we want to find v0, the voltage across the 10 k-ohm resistor.
Well use Ohms Law and Kirchhoffs current and voltage laws (KCL and KVL).
First label the all currents and voltages in the circuit.
We
Here we want to find v1, the voltage across the 10 k-ohm resistor.
Well use Ohms Law and Kirchhoffs current and voltage laws (KCL and KVL).
First label the all currents and voltages in the circuit.
We
Quiz 2
- ECE161B
Winter 15, Song
Consider the following multi-rate system where H (z) is an ideal lowpass filter with a
cutoff frequency at / 2 .
a. (50pt) Use the noble identities (cascade equivalenc
FINAL
ECE 161A, Fall 2012
Some Guidelines for the Fall
Please sit as far apart as possible.
Please keep your answer sheet under your control at all times.
Please show your work as it may enable you
Problem
A stable system with system function 11(2) has the pole-zero diagram shown in Figure 1. It can
be represented as the cascade of a stable minimum-phase system H min (2:) and a stable all-pass
s
1
Midterm Solution
1)
a)
X[k] =
=
=
=
=
N
1
X
2
g[n]h[n]eJ N nk
n=0
N
1 N
1
X
X
1
N
1
N
1
N
1
N
2
n=0 m=0
N
1
N
1
X
X
G[m]
m=0
N
1
X
m=0
N
1
X
2
G[m]eJ N mn h[n]eJ N nk
G[m]
n=0
N
1
X
2
h[n]eJ N n(km)