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Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
LECTURE 1:
M-D SIGNALS AND TRANSFORMS
• M-D Signals
– Finite-Extent Signals and Periodic Signals
– Symmetric Signals
– Special signals
• M-D Transforms
–
–
–
–
–
MD-FT for Continuous Signals
MD-FT for Discrete Signals
MD-DFT
MD-DCT
MD-Wavelet
Chapter 1 Multi-dimensional Signals and Systems
1
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Finite-Extent Signals
Finite-extent signals are defined over a finite support.
• Quarter-plane (QP) support
• Half-plane (HP) support
• Non-symmetric half-plane (NSHP) support
• Wedge support
n2
n1
Chapter 1 Multi-dimensional Signals and Systems
2
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Periodic Signals
̃
• Definition:
̃
|
• Arbitrary periodicity
• Rectangular-periodicity:
Chapter 1 Multi-dimensional Signals and Systems
and
diagonal
3
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Periodic Extension of Finite-Extent Signals
• Finite-extent signals and periodic signals are isomorphic to
each other.
– Given a periodic signal, the main period is finite-extent
– Given a finite extent signal, we can always define its
periodic extension
̃
,
Chapter 1 Multi-dimensional Signals and Systems
,
4
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Symmetric Signals
• Two-fold (NSHP) symmetry
,
• Four-fold (QP) symmetry
,
,
,
,
• Circular symmetry
– A signal
,
is circularly symmetric if it is only a
from the origin. Circular
function of distance
symmetry implies four-fold symmetry.
Chapter 1 Multi-dimensional Signals and Systems
5
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Separable Signals
• An MD signal (function) is separable if
, ,⋯,
⋯
,
can be represented by
• A finite support 2D signal
a matrix . If the signal is separable, then the matrix can
be written as the outer product
, where the
vectors and denote samples of 1D signals
and
, respectively.
• While a general
matrix has
degrees of
freedom, the outer product has
degrees of
freedom.
Chapter 1 Multi-dimensional Signals and Systems
6
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Special Signals: 2D Kronecker Delta
• MD Kronecker-delta signal
,
1
0
,⋯,
⋯
0
otherwise
• 2D discrete time signals can be expressed as shifted and
weighted sum of Kronecker delta signals.
s(1,1)
s(0,1)
s(3,1)
s(0,0)
n2
s(4,2)
s(1,0)
n1
Chapter 1 Multi-dimensional Signals and Systems
7
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Special Signals: 2D Spatial Frequency Patterns
• Horizontal pattern
,
cos
,
cos
• Vertical pattern
• 45-degree orientation
,
• Complex exponential
cos
– A 2D discrete complex exponential signal
,
is rectangularly periodic in
where
and
,
,
,
and
with period
,
if
and
are unitless integers, and the units of
are radians.
Chapter 1 Multi-dimensional Signals and Systems
8
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
2D Fourier Transform of Continuous Signals
• Forward transform
,
,
,
• Inverse transform
1
,
2
,
,
• MD Fourier transform is complex
,
,
,
• Frequency variables
2 ,
cycles/mm,
Chapter 1 Multi-dimensional Signals and Systems
,
,
radians/mm
9
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Convergence of the Fourier Transform
The 2D signal
,
may have an infinite extent.
• Uniform convergence: The double integral converges uniformly and is a
continuous function of
and
if
∬
•
•
,
∞; i.e.,
,
is absolutely integrable.
,
exists but has discontinuities, then a
Mean-square convergence: If
weaker form of convergence applies. For example,
,
is not absolutely integrable, but its Fourier transform
sin ⁄
sin ⁄
converges in the mean square sense. We observe the Gibbs effect around points
of discontinuity.
Generalized convergence: In some cases, neither uniform nor mean-square
convergence applies, but
,
may still be defined using the Dirac delta
function
,
. For example,
,
1 for all
,
, is not
absolutely integrable, but its Fourier transform is defined in the generalized
,
,
.
sense as
Chapter 1 Multi-dimensional Signals and Systems
10
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
2D Fourier Transform: Coordinate Transforms
• Let
,
,
denote an affine transformation of coordinates. If
,
is the
2D Fourier transform of
,
, then the 2D Fourier transform
,
is given by
of
1
,
,
1,
,
• Translation:
0,
• Rotation:
0.
,
sin,; where
cos,
1.
,
cos
Chapter 1 Multi-dimensional Signals and Systems
sin ,
sin
cos
11
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Projection Slice Theorem
• Let
,
have Fourier transform
denote the Radon transform of
,
cos
which projects
Then,
,
sin ,
,
, and
defined by
sin
cos
to a line through the origin with angle .
cos , sin
denotes the 1D Fourier transform of
for
where
each angle .
• The projection-slice theorem is fundamental to how several
medical imaging modalities, e.g., computer tomography, works.
Chapter 1 Multi-dimensional Signals and Systems
12
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Projection Slice Thm. (cont’d)
• Proof for the case
Projection of
,
0
on the horizontal axis is defined by
,
Taking the 1D Fourier transform of the projection
yields
,
,
Chapter 1 Multi-dimensional Signals and Systems
,0
13
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
2D Fourier Transform of Discrete Signals
• Forward transform is periodic with period 2
,
,
• Inverse transform
1
,
2
• Properties:
–
,
real implies
–
,
,
2
,
,
has conjugate symmetry.
is two‐fold symmetric implies
∗
,
→
,
Chapter 1 Multi-dimensional Signals and Systems
,
∗
is real
,
14
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
2D Discrete Fourier Transform
• 2D DFT can be obtained by sampling
,
of a finite-extent
signal
,
or by computing the Fourier series coefficients of
the periodic extension ̃ ,
,
,
• Normalized frequency variables
2
and
∆
∆
2
Chapter 1 Multi-dimensional Signals and Systems
and
2
∆
∆
2
15
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Computation of 2D Discrete Fourier Transform
• Since 2D complex exponentials are separable, 2D DFT can be
computed as a cascade of two 1D DFTs, first on the rows of
,
, then on the columns of
,
as
,
,
where
,
is the 1D DFT over the row
Chapter 1 Multi-dimensional Signals and Systems
,
of the image.
16
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Computation of 2D IDFT
• Inverse 2D DFT can be computed using the forward FFT algorithm
by first conjugating
,
, then computing 2D forward DFT,
and again taking the complex conjugate of the result, since we have
1
,
1
Chapter 1 Multi-dimensional Signals and Systems
,
∗
∗
,
17
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Properties of 2D Discrete Fourier Transform
• 2D DFT is rectangularly periodic
,
,
,
•
,
,
real implies
for all
is Hermitian symmetry
| 0,0 |
| 1,0 |
| 2,0 |
| 3,0 |
2,0
1,0
| 0,1 |
| 1,1 |
| 2,1 |
| 3,1 |
| 4,1 |
| 5,1 |
| 0,2 |
| 1,2 |
| 2,2 |
| 3,2 |
| 4,2 |
| 5,2 |
| 0,3 |
| 1,3 |
| 2,3 |
| 3,3 |
2,3
1,3
0,2
5,2
4,2
3,2
2,2
0,1
5,1
4,1
3,1
2,1
0
1,0
2,0
0
0,1
1,1
2,1
3,1
4,1
5,1
0,2
1,2
2,2
3,2
4,2
5,2
0
1,3
2,3
0
2,3
1,3
2,0
1,2
1,1
1,0
0,2
5,2
4,2
3,2
2,2
1,2
0,1
5,1
4,1
3,1
2,1
1,1
Chapter 1 Multi-dimensional Signals and Systems
,
18
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
Properties of 2D DFT (cont’d)
• Circular Shift
,
↔
,
• Circular Convolution
,
⊛⊛
,
↔
,
,
• Parseval’s Theorem:
,
Chapter 1 Multi-dimensional Signals and Systems
1
,
19
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
2D Discrete Cosine Transform
• Symmetric extension for type II DCT is given by
2
1
,
0
• The algorithm to compute -point DCT
using 2 -point FFT is as follows:
of
2
-point signal
1) Form 2 -point symmetrically extended signal
.
2) Compute
,
0, … , 2
1, the 2 -point DFT of
3)
⁄
,
0, … ,
1
.
1, where
• The high frequency coefficients of the DCT contains less energy
compared with those of the DFT due to symmetric extension.
Chapter 1 Multi-dimensional Signals and Systems
20
Prof. A. Murat Tekalp
Digital Video Processing, 2E, Prentice Hall, 2015
2D Discrete Wavelet Transform (DWT)
• 2D DWT is a multi-scale image representation.
• Discussion of 2D DWT is delayed until we study FIR filtering
and multi-scale image representations in Chapter 3.
Chapter 1 Multi-dimensional Signals and Systems
21
@Copyright 2015 Prof. A. Murat Tekalp
Digital Video Processing
Fall 2015
Display of 2D Signals and Transforms
•
Two-dimensional functions can be displayed as gray-scale plots, isometric
(surface) plots, or contour plots, which are supported by MATLAB.
•
Appropriate scaling is important for gray-scale plots, e.g., to display 2-D Fourier
transform of images, or to compare multiple images. Common approaches for
scaling are linear min/max scaling and nonlinear scaling. For example, log scaling
for Fourier magnitude is given by
•
•
,
log 1
,
where D() denotes the display image and F() denotes the actual Fourier transform.
Surface plots give the appearance of 3D drawing. A wire mesh or a shaded solid
can represent the surface. Some portions of the surface may be occluded. This is
very useful to display point spread functions and frequency response of systems.
In contour plots, all points that have a specific value are connected to form a
continuous line. This type of plot is useful locating minima and maxima of twodimensional functions.
Chapter 1 Multi-dimensional Signals and Systems
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