DIGITAL IMAGE
PROCESSING
Instructors:
Dr J. Shanbehzadeh
Shanbehzadeh@gmail.com
M.Gholizadeh
mhdgholizadeh@gmail.com
( J.Shanbehzadeh M.Gholizadeh )
DIGITAL IMAGE
PROCESSING
Instructors:
Dr J. Shanbehzadeh
Shanbehzadeh@gmail.com
M.Gholizadeh
mhdgholizadeh@gmail.com
( J.Shanbehzadeh M.Gholizadeh )
Road map of chapter 7
7.1
7.2
7.7
7.3
7.4
Wavelet
Transform
in Two
One
The
Fast
Wavelet Expansions
Transform
Wavelet
Packets
Multi
Resolution
Background
Dimensions
Dimension
( J.Shanbehzadeh M.Gholizadeh )
7.5
7.6
7.1 Background
7.2 Multi Resolution Expansions
7.3 Wavelet Transform in One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two Dimensions
7.6 Wavelet Packets
Wavelets and Multi-resolution
Processing

Preview

What is multi-resolution?
- unifies techniques from a variety of disciplines,including subband coding
from signal processing, quadrature
mirror filtering from digital speech recognition, and pyramidal image
processing.
- features that might go undetected at one resolution may be easy to detect at
another.
( J.Shanbehzadeh M.Gholizadeh )
The difference between Fourier
transform and Wavelet transform
1) Fourier transform’ s basis functions are sinusoids, wavelet transforms are based
on small waves, called wavelets, of varying frequency and limited duration.
2) Fourier transforms, provide only frequency information and temporal
information is lost in the transformation process.
( J.Shanbehzadeh M.Gholizadeh )
Background
If both small and large objects, or low and high contrast objects are present need
multiresolution
Examine an object --Depending on the size or contrast of the object  choose the
resolution(high , low)
Local histogram variations (Fig. 7.1)
( J.Shanbehzadeh M.Gholizadeh )
background
local histograms can vary from one part of an image to another
making statistical modeling over the span of an entire image is a difficult, or
impossible task.
( J. Shanbehzadeh M.Gholizadeh )
Background
7.1 Background
7.2 Multi Resolution
Expansions
Image Pyramids
pyramids
Subband Coding
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
The Haar Transform
Image Pyramids
7.1 Background
What is an image pyramid?
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
A powerful , simple structure for representing images at
more than one resolution.
an image pyramid is a collection of decreasing resolution
images arranged in the shape of a pyramid .
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
9
( J. Shanbehzadeh M.Gholizadeh )
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J. Shanbehzadeh M.Gholizadeh )
(a) : The base of the pyramid contains a high-resolution
representation of the image being
Processed; the apex contains a low-resolution approximation .
As you move up the pyramid, both size and resolution
decrease.
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
provides the
images
needed to
build an
approximation
pyramid
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
is used to build a
complementary prediction residual
pyramid.
prediction residual pyramids
contain only one reducedresolution approximation of the
input image
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J. Shanbehzadeh M.Gholizadeh )
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J. Shanbehzadeh M.Gholizadeh )
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Image Pyramids
•
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
A P+I level pyramid is built by executing the operations in the block
diagram P times
–
first iteration produces the level J-1 approximation and level J
residual results
–
each pass is composed of three steps (Fig. 7.2(b))

Step 1: compute a reduced-resolution approximation of the
input image:filtering and down-sampling

Mean pyramid, low-pass Gaussian filter based on
Gaussian pyramid, no filtering (i.e.sub-sampling pyramid)

If we compute without filtering, alias can become
pronounced

Step 2
1. up-sample the o/p of the step (a)-again by a factor of 2. filter-interpolate intensities between the pixels of the step 1



( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Create a prediction image
Determines how accurately approximate the input by using
interpolation
If we delete interpolation filter, blocky effect is inevitable
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
 Step3 : compute the difference between the prediction of
step2 and the input to step 1 (prediction residual)
 Predict residual of level J


7.3 Wavelet Transform in
One Dimension
Can be used to reconstruct the original image
Can be used to generate the corresponding approximation
pyramid including the original image without quantization
error
 level j-1 approximation can be used to populate the
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
approximation pyramid
 coarse to fine strategy
 High resolution pyramid—used for analysis of large
structure or overall image context
 Low resolution pyramid —analyzing individual object
characteristics
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
 the level j prediction residual outputs are placed in the
prediction residual pyramid
 Ex. Fig. 7.7 (P=7)
 Approximation pyramid--Gaussian pyramid (5x5
low-pass Gaussian kernel)
 Prediction residual--Laplacian pyramid
 64x64 Laplacian pyramid predict the Gaussian
pyramid’s level 7 prediction residual
 First order statistics of the pyramid are highly
peaked around zero
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
the lower-resolution levels of a
pyramid can be used for
the analysis of large structures or
overall image context
Background
Image Pyramids
7.1 Background
7.2 Multi Resolution
Expansions
Subband Coding
7.3 Wavelet Transform in
One Dimension
The Haar Transform
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Subband Coding
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Definition :
in subband coding :an image is decomposed into a set of band
limited components, called subbands. The decomposition is
performed so that the subbands can be reassembled to reconstruct
the original image without error.
 A filter bank is a collection of two or more filters.
Subband Coding
The goal in subband coding is to select h0(n),h1(n),g0(n),g1(n) so that x(n) = x’(n)
.
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
filters go(n) and g1(n) combine y0(n) and y1 (n)
to produce x’(n).
( J.Shanbehzadeh M.Gholizadeh )
Subband Coding

7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform





7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
An image is decomposed into a set of band-limited
component sub-bands, which can be reassemble to
reconstruct the original image
Each sub-band is generated by band-pass filtering its
I/p
the sub-band can be down sampled without loss of
information
Reconstruction of the original image is accomplished
by sampling, filtering, and summing the individual
sub-band
The principal components of a two-band sub-band
coding and decoding system (Fig. 7.4)
The output sequence is formed through the
decomposition of x(n) into y0(n) and y1(n) via
analysis filter h0(n) and h1(n),and subsequent
recombination via synthesis filters g0(n) and g1(n)
Subband Coding
For perfect reconstruction,
the impulse responses of the
synthesis and analysis filters
must be related in one of the
following two ways:
 Bio-orthogonal- filter bank satisfying the conditions
 Filter response of two-band, real coefficient, perfect reconstruction filter bank are subject to
bio-orthogonality constraints
 Orthonormal
28
Subband Coding
1-D orthonormal and biorthogonal filters can be used as 2-D separable
filters for the processing of images.
approximation
vertical detail
horizontal detail
diagonal detail
the separable filters are first applied in one dimension (e.g., vertically) and then in the
other(e.g..horizontally) .
 Down sampling is performed in two stages-once before the second filtering operation to
reduce the overall number of computations .
29
( J.Shanbehzadeh M.Gholizadeh )
Subband Coding
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
It is easy to show numerically
that the filters are both biorthogonal and orthonormal.
30
As a result, it supports error-free reconstruction of the decomposed input.
Subband Coding
approximation
vertical detail
•visual effects of aliasing that are present in Figs. 7.7(b)
and c.
• The wavy lines in the window area are due to the downsampling of a barely discernable window screen in Fig. 7.1.
• Despite the aliasing, the original image can be
reconstructed from the subbands in Fig. 7.7 without
error.
horizontal detail
diagonal detail
( J.Shanbehzadeh M.Gholizadeh )
31
Background
7.1 Background
Image Pyramids
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Subband Coding
The Haar Transform
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
 Basic functions are the oldest and simplest known orthnormal
wavelet
 Separable and symmetric and can be expressed in matrix form
T=HFH
where F is an N * N image matrix, H is an N X N Haar transformation
matrix, and T is the resulting N X N transform
The Harr basic functions are :
 z€[0 1],k=0,1,2,…,N,N=2^n , k=2^p+q-1,0≤p≤n-1
0 or 1
p=0
q=
0≤q≤2^p p≠0
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
( J.Shanbehzadeh M.Gholizadeh )
The Harr transform
( J.Shanbehzadeh M.Gholizadeh )
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Problms
 Problem 7.1
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
 Problem 7.2
 Problem 7.7
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
 Problem 7.4
7.6 Wavelet Packets
 Problem 7.5
 Due Date Friday 21/12/88
( J.Shanbehzadeh M.Gholizadeh )
Why is orthogonality useful
x  1a1   2a 2
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
x  1 1
T
a 1  2 1
T
a 2   1 2
T
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
1  x, a1 / a1 , a1  3 / 5
 2  x, a 2 / a 2 , a 2  1 / 5
Orthonormal bases further simplify the 
computation
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Ortho v. Non-Ortho Basis
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Dual Basis
a 1  2 1
T
a 2  1 2
T
a1 , b1
a 2 , b2
a1 , b 2
a 2 , b1
x  1 1
T
1
1
0
0
1  x, b1 / a1 , b1  1 / 3
 2  x, b2 / a 2 , b2  1 / 3
Dual Bases
b1  2 / 3  1 / 3
T
b 2   1 / 3 2 / 3
T
( J.Shanbehzadeh M.Gholizadeh )
x  1a1   2a 2
a1-a2 and b1-b2 are
biorthogonal
Dual Basis (cont)
a1  1  1 2
T
a 2  0 1 0
T
b1  1 0 0
Verify
duality !
T
b 2  1 1 0
T
 Dual basis may generate different spaces
 Here: a1-a2 and b1-b2 generate two different 2D subspaces in
Euclidean 7space.
 Semiorthogonal:
 For dual basis that generates the same subspace
 Orthogonal:
 Primal and dual are the same bases
( J.Shanbehzadeh M.Gholizadeh )
Multi Resolution Expansions
7.1 Background
Series Expansions
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Scaling Functions
Wavelet Functions
series expansions
 A signal f(x) can be analyzed as a linear combination of expansion function
real-valued expansion functions
or basis function
real-valued expansion coefficients
closed span of the expansion set
Inner product
( J.Shanbehzadeh M.Gholizadeh )
expansion set
dual functions
Multi Resolution Expansions
Series Expansions
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Series Expansions
Scaling Functions
Wavelet Functions
Scaling functions
Scaling functions
- Consider the set of expansion functions composed of integer translations and binary
scaling of the real, square-integrable function φ(x),this is the set {φj,k(x)}, where
k
j
2^(j/2)
the position of φj,k(x)along the x-axis
width of φj,k(x)
controls the amplitude of the function.
Because the shape of φj,k(x)changes with j, φ(x) is called a scaling function
For J=J0
( J.Shanbehzadeh M.Gholizadeh )
Scaling functions
(a-d) :four of the many expansion
functions that can be generated
by substituting this pulse-shaped
scaling function into
e: shows a member of subspace V1.
It does not belong to V0, because
the V0 expansion functions in(a,b)
are too coarse to represent it.
f: the decomposition ofΦ0,0 (x) as a sum
of V1 expansion functions.
( J.Shanbehzadeh M.Gholizadeh )
Scaling functions
 four fundamental requirements of multiresolution analysis :
 1) The scaling function is orthogonal to its integer translates.
for Haar function, it has a value of 1, its integer translates are 0, so that the
product of the two is 0 .
 2) The subspaces spanned by the scaling function at low scales are nested
within those spanned at higher scales.
 7) The only function that is common to all Vj is F(X) = 0.If we consider
the coarsest possible expansion functions ( j = -∞), the only representable
function is the function of no information.
 4) Any function can be represented with arbitrary precision.
all measurable, square-integrable functions can be represented by the
scaling functions in the limit as j∞.
( J.Shanbehzadeh M.Gholizadeh )
Scaling Functions
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
subspaces containing high-resolution
functions must also contain all lower
resolution functions.
( J.Shanbehzadeh M.Gholizadeh )
Multi resolution expansions
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
Series Expansions
7.4 The Fast Wavelet Transform
Scaling Functions
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Wavelet Functions
Wavelet functions
wavelet function ψ(x) that, together with its integer translates and binary scaling, spans
the difference between any two adjacent scaling subspaces. Vj and Vj+1.
The set {ψj,k(x)} of wavelets
for all k€Z that span the Wj spaces in the figure. As with scaling functions, we write

and if f(x)€Wj
Since wavelet spaces reside within the spaces spanned by the next higher
solution scaling functions , any wavelet function can be expressed as a weighted sum
shifted, double-resolution scaling functions. we can write :
 hψ (n) : wavelet function coefficients
( J.Shanbehzadeh M.Gholizadeh )
Wavelet functions
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
The scaling and wavelet function subspaces in Fig. 7.11
are related by
( J.Shanbehzadeh M.Gholizadeh )

union of subspaces
Wavelet functions
generate the universe of scaled
and translated Haar wavelets
•Waveletψ1,0(x) for space W1 is narrower than
ψ0,2(x) for W0;
it can be used to represent finer detail.
•d : shows a function of subspace V1
that is not in subspace V0.
•(e-f) divide f(x) in a manner similar to a lowpass
and highpass filter
( J.Shanbehzadeh M.Gholizadeh )
The Wavelet series Expansions
• defining the wavelet series expansion of function f(x) relative to wavelet ψ(x) and
scaling functionφ(x) : (j0 is an arbitrary starting scale )
The c j0(k) are normally called approximation and/or
scaling coefficients
the dj(k) are referred to as
detail and/or wavelet
coefficients.
For each higher scale j≥j0 in the second sum, a finer
resolution function —a sum of wavelets-is
added to the approximation63to provide increasing detail.
( J.Shanbehzadeh M.Gholizadeh )
Problems
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets










7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.17
7.14
Due date Monday 25/12/88
64
( J.Shanbehzadeh M.Gholizadeh )
Multi Resolution Expansions
The
The Wavelet Series
Series Expansions
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
The Discrete Wavelet Transform
The Continuous Wavelet Transform
The Wavelet series Expansions
66
( J.Shanbehzadeh M.Gholizadeh )
Multi Resolution Expansions
The Wavelet Series Expansions
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
The
TheDiscrete
DiscreteWavelet
WaveletTransform
Transform
The Continuous Wavelet Transform
The Discrete Wavelet Transform
Like the Fourier series expansion, the wavelet series expansion
maps a function of a continuous variable into a sequence of coefficients.
If the function being expanded is discrete ,the resulting coefficients are called the
discrete wavelet transform (DWT) .
 the series expansion becomes the DWT transform pair: (x=0,….,M-1)
TheWφ(jo, k) and Wψ (j, k) correspond to the cj0(k) and dj(k) of the wavelet series
expansion in the previous section.
Inverse DWT
1/√M is normalizing factor
( J.Shanbehzadeh M.Gholizadeh )
69
( J.Shanbehzadeh M.Gholizadeh )
EXAMPLE7.8
( J.Shanbehzadeh M.Gholizadeh )
Multi Resolution Expansions
7.1 Background
7.2 Multi Resolution
Expansions
Series Expansions
7.3 Wavelet Transform in
One Dimension
Scaling Functions
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Wavelet Functions
The Continuous Wavelet
Transform
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
Inverse CWT
( J.Shanbehzadeh M.Gholizadeh )
• The natural extension of the discrete wavelet transform
Transforms a continuous function into a highly redundant function of
two continuous
variables(translation and scale )
W a, b   


f t 
 a ,b t  
1
a
t b
dt
 a 
 *
t b


a  a 
1
Fourier spectrum
Mexican hat
b:reveals the close connection
between scaled wavelets and Fourier
frequency bands. The spectrum
contains
Two peaks that correspond two
Gaussian-like perturbations of the
function.
c:a portion of the CWT of(a)
relative to the Mexican hat
wavelet.Unlike(b), it provides
both spatial and frequency
information
( J.Shanbehzadeh M.Gholizadeh )
d: the absolute value of the transform |Wψ(s,τ ) | is displayed as
intensities between black and white
7.4 the fast wavelet
transform
74
( J.Shanbehzadeh M.Gholizadeh )
Multiresolution Refinement
 scaling x by 2^j. translating it by k, and letting m = 2k + n gives
75
( J.Shanbehzadeh M.Gholizadeh )
Multiresolution Refinement

interchanging the sum and integral
If j0 =j+1
Like above
 finally we have
76
( J.Shanbehzadeh M.Gholizadeh )
FWT
7.1 Background
a computationally efficient implementation of
the discrete wavelet transform (DWT) .
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
the filter bank can be "iterated" to create multistage structures
for computing DWT coefficients at two or more successive scales.
( J.Shanbehzadeh M.Gholizadeh )
splits the original function into
a lowpass, approximation component
and a highpass, detail component
splits the spectrum and subspace, the lower
half-band, into quarter-band subspaces.
( J.Shanbehzadeh M.Gholizadeh )
a two-stage filter bank for generating the coefficients at the two highest
scales of the transform.The highest scale coefficients are assumed to
be samples of the function itself.
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
Fast wavelet Transform
synthesis filter bank
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
Is identical to the synthesis portion of the two-band subband coding and decoding system in Fig. 7.4(a).
( J.Shanbehzadeh M.Gholizadeh )
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 Wavelet Transform in Two
Dimensions
7.6 Wavelet Packets
( J.Shanbehzadeh M.Gholizadeh )
a two-scale structure for computing the final two scales of
a (FWT)^(-1)
reconstruction is depicted. This can be extended to any
number of scales
and quarantees perfect reconstruction of sequence f(n).
Negative indices n<0
shows the sequences that result from the required FWT convolutions and downsamplings.
Function f(n) itself is the scaling (approximation) input to the leftmost filter bank.
( J.Shanbehzadeh M.Gholizadeh )
7.1 Background
7.2 Multi Resolution
Expansions
7.3 Wavelet Transform in
One Dimension
7.4 The Fast Wavelet Transform
7.5 indices
Wavelet Transform
in Two
Negative
n<0
Dimensions
7.6 Wavelet Packets
illustrates the process for the sequence
considered
in Example 7.10.
( J.Shanbehzadeh M.Gholizadeh )
an impulse function basis
sinusoidal (FFT) basis
FWT basis
low frequencies
the tiles are shorter
but are wider
high frequencies
tile width is smaller
height is greater .
A: events occur but provides no frequency information . Thus, to represent a single frequency sinusoid as an
expansion using impulse basis functions, every basis function is required.
B: the frequencies in events that occur over long periods but provides no time resolution .
Thus, the single frequency sinusoid that was represented by an infinite number of impulse basis
functions can be represented as an expansion involving one sinusoidal basis function.
The time and frequency resolution of the FWT tiles in (c) vary, but the area of each tile (rectangle) is the same.
Thus,the FWT basis functions provide a compromise between the two limiting cases(a) and (b).
( J.Shanbehzadeh M.Gholizadeh )
7.5Wavelet Transforms in Two
Dimensions
( J.Shanbehzadeh M.Gholizadeh )
Wavelet Transforms in Two
Dimensions
 a two-dimensional scaling function, φ(x, y), and three two dimensional wavelets,
ψH (x, y),ψV(x,y),ψD(x,y), are required. Each is the product of two onedimensional functions.
 the separable scaling function
measures variations along columns (for example, horizontal edges)
responds to variations along rows (like vertical edges)
corresponds to variations along diagonals
86
( J.Shanbehzadeh M.Gholizadeh )
Wavelet Transforms in Two
Dimensions
 scaled and translated basis functions:
 The discrete wavelet transform of image f(x. y) of size M X N is then
coefficients define an approximation
of f(x, y) at scale j0
coefficients add horizontal, vertical, and diagonal details for scales j≥j0
Inverse DWT
87
( J.Shanbehzadeh M.Gholizadeh )
a:Convolving its rows with hφ (-n) and hψ (-n) and downsampling its columns, we get two
subimages whose horizontal resolutions are reduced by a factor of 2. Both subimages are then
filtered columnwise and downsampled to yield four quarter-size output subimages .
The highpass or detail component  the image's high-frequency information
with vertical orientation;
The lowpass, approximation component  low-frequency, vertical information.
b:These subimages, are the inner products of f(x, y) and the two-dimensional scaling
and wavelet functions followed by downsampling by two in each dimension.
c:the synthesis filter bank reverses the process. At each iteration, four scale j
approximation and detail subimages are upsampled and convolved with two one-dimensional
filters one operating on the subimages' columns and the other on its rows. Addition of the
results yields the scale j + 1approximation, and the process is repeated until the original
image is reconstructed.
( J.Shanbehzadeh M.Gholizadeh )
Each pass through the filter bank produced four
quarter-size output images that were substituted
for the input from which they were derived.
2-Dfilter bank of 7.24(a) and the decomposition
filters shown in Figs. 7.26(a,b) were used to
generate all three results.
d: is the three-scale FWT that resulted when
the subimage from the upper-left-hand corner
of(c) was used as the filter bank input .
A similar process for generating the two-scale
FWT in (c), but the input to the filter bank was
changed to the quarter-size approximation subimage from the upper-left-hand corner of (b).
computer-generated image consisting of
2-D sine-like pulses on a black background
( J.Shanbehzadeh M.Gholizadeh )
(b) To compute this transform, the original image was
used as the input to the filter bank of 7.24(a) . The four
resulting quarter-size decomposition outputs (i.e.,on the
mechanics of the the approximation and horizontal,
vertical, and diagonal details)were then arranged in
accordance with Fig. 7.24(b) to produce the image in
7.25(b).
Next example
The decomposition filters usd in the preceding example are part of a well known
family of wavelets called symlets, short for "symmetrical wavelets.“
 (e) and (f) show the fourth-ordervalues. 1-D symlets (wavelet and scaling functions)
 7.26(a) through (d) show the corresponding decomposition and reconstruction filters.
Figure 7.26(g), a low-resolution graphic depiction of wavelet ψV(x, y), is provided
as an illustration of how a one-dimensional scaling and wavelet function can combine
to form a separable, two-dimensional wavelet .
( J.Shanbehzadeh M.Gholizadeh )
a:Convolving its rows with hφ (-n) and hψ (-n) and downsampling its columns, we get two
subimages whose horizontal resolutions are reduced by a factor of 2. Both subimages are then
filtered columnwise and downsampled to yield four quarter-size output subimages .
The highpass or detail component  the image's high-frequency information
with vertical orientation;
The lowpass, approximation component  low-frequency, vertical information.
b:These subimages, are the inner products of f(x, y) and the two-dimensional scaling
and wavelet functions followed by downsampling by two in each dimension.
c:the synthesis filter bank reverses the process. At each iteration, four scale j
approximation and detail subimages are upsampled and convolved with two one-dimensional
filters one operating on the subimages' columns and the other on its rows. Addition of the
results yields the scale j + 1approximation, and the process is repeated until the original
image is reconstructed.
( J.Shanbehzadeh M.Gholizadeh )
The coefficients of lowpass reconstruction filter go(n) = hφ(n) for 0 < n < 7 .
 The coefficients of the remaining orthonormal filters are obtained using
The coefficients of lowpass reconstruction filter go(n) = hφ(n) for 0 < n < 7 .
 The coefficients of the remaining orthonormal filters are obtained using
( J.Shanbehzadeh M.Gholizadeh )




As in the Fourier domain, the basic approach is to
Step 1. Compute a 2-D wavelet transform of an image.
Step 2. Alter the transform.
Step 7. Compute the inverse transform.


a: the lowest scale approximation component of the discrete wavelet transform shown
in Fig. 7.25(c) has been eliminated by setting its values to zero.
 7.27(b) shows, the net effect of computing the inverse wavelet transform using these

modified coefficients is edge enhancement.


Note how well the transitions between signal and background are delineated, despite
the fact that they are relatively soft, sinusoidal transitions.


By zeroing the horizontal details as well—see Figs. 7.27(c) and (d)—we can isolate
the vertical edges .
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
The coefficients of lowpass reconstruction filter go(n) = hφ(n) for 0 < n < 7 .
 The coefficients of the remaining orthonormal filters are obtained using
( J.Shanbehzadeh M.Gholizadeh )
Next Example




As in the Fourier domain, the basic approach is to
Step 1. Compute a 2-D wavelet transform of an image.
Step 2. Alter the transform.
Step 7. Compute the inverse transform.


a: the lowest scale approximation component of the discrete wavelet transform shown
in Fig. 7.25(c) has been eliminated by setting its values to zero.
 7.27(b) shows, the net effect of computing the inverse wavelet transform using these

modified coefficients is edge enhancement.


Note how well the transitions between signal and background are delineated, despite
the fact that they are relatively soft, sinusoidal transitions.


By zeroing the horizontal details as well—see Figs. 7.27(c) and (d)—we can isolate
the vertical edges .
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
Next Example
general wavelet-based procedure for denoising the image :
Step 1. Choose a wavelet (Haar. symlet ) and number of levels (scales), P, for
the
decomposition. Then compute the FWT of the noisyimage.
Step 2. Threshold the detail coefficients. That is, select and apply a threshold to the detail coefficients .Soft thresholding eliminates the Step 7. Compute the inverse wavelet transform (i.e., perform a wavelet
discontinuity
reconstruction)
(at the threshold) that is inherent in hard thresholding.
using the original approximation coefficients at level J - P and
the
modified detail coefficients for levels J — 1 to J — P.
Thresholding
hard thresholding,means setting to zero
the elements whose absolute values are
lower than the threshold
( J.Shanbehzadeh M.Gholizadeh )
soft thresholding, involves first setting to zero the
elements whose absolute values are lower than the
threshold and then scaling the nonzero coefficients toward
zero
(b) :result of performing these
operations with fourth-order symlets, two
scales (P = 2), and
a global threshold(determined interactively)
the reduction in noise and blurring
of image edges. This loss of edge
detail is reduced significantly in (c)
f) shows the information that is lost.
note the increase in edge information
in(f) and the corresponding decrease
in edge detail in (e).
generated by simply zeroing
the highest-resolution detail
coefficients and reconstructing
e) Reconstruction of the DWT in which the details at
both levels of the two-scale transform have been zeroed;
shows the information that is lost in the process. which was generated
by computing the inverse FWT of the two-scale transform with all but the
highest-resolution detail coefficients zeroed
( J.Shanbehzadeh M.Gholizadeh )
7.6 Wavelet Packets
Wavelet Packets
Wavelet Packets
If we want greater control over the partitioning of the time-frequency plane , the FWT
must be generalized to yield a more flexible decomposition .
The cost
increase in computational complexity from O(M) for
the FWT to O(Mlog M) for a wavelet packet.
103
 Figure 7.29(a) links the appropriate FWT scaling and wavelet coefficients to its nodes.
 The root node is assigned the highest-scale approximation coefficients, which are
 samples of the function itself, while the leaves inherit the transform‘s approximation
 and detail coefficient outputs. The lone intermediate node, is a filter bank
 approximation that is ultimately filtered to become two leaf nodes .


replace the generating coefficients in Fig. 7.29(a) by the corresponding subspace.
The result is the subspace analysis tree of Fig. 7.29( b).
( J.Shanbehzadeh M.Gholizadeh )
the block diagram of (a) is labeled to resemble the analysis tree in (b). while
Ihe output of the upper-left filter and subsampler is. to be accurate,
Wψ(J - 1, n), it has been labeled WJ-1. This subspace corresponds to the
upper-right leaf of the associated analysis tree, as well as the right most
(widest bandwidth) segment of the corresponding frequency spectrum.
Analysis trees provide a compact
and informative way of representing
multiscale wavelet transforms .
( J.Shanbehzadeh M.Gholizadeh )
( J.Shanbehzadeh M.Gholizadeh )
 the three-scale FWT analysis tree of 70(b) becomes the three-scale wavelet packet tree of 71.
 Note the additional subscripting :

The first subscript of a double-subscripted node identifies the scale of the FWT parent

node from which it descended. The second(a variable length string of As and Ds)

encodes the path from the parent to the node. An A designates approximation filtering,

while a D indicates detail filtering.
( J.Shanbehzadeh M.Gholizadeh )
7.72(a-b) are the filter bank and spectrum splitting characteristics of the analysis tree in7.71. Note that the "naturally ordered“
output of the filter bank in (a) have been reordered based on frequency content in (b)
( J.Shanbehzadeh M.Gholizadeh )
consider the two-dimensional, four-band filter bank of Fig. 7.24(a). As in the one
dimensional case, it can be "iterated" to generate P scale transforms for scales j = J- 1,... ,J- P,
with Wφ(J.m. n) = f(m, n). The spectrum resulting from the first iteration is shown in 7.74(a).
 Note that it divides the frequency plane into four equal areas.The low-frequency quarterband in the center of the plane coincides with transform coefficients Wφ(J – 1,m, n) and
scaling space Vj-1
 b: shows the resulting four-band.single-scale quaternary FWT analysis tree.the superscripts
that link the wavelet subspace designations to their transform coefficient counterparts.
( J.Shanbehzadeh M.Gholizadeh )
Like its one-dimensional , the first subscript of every node that is a descendant of a
conventional FWT detail node is the scale of that parent detail node.
The second subscript(a variable length string of As, Hs,Vs,Ds)encodes the path from
the parent to the node under consideration.
( J.Shanbehzadeh M.Gholizadeh )
efficient algorithm for finding
optimal decompositions with
respect to application specific
criteria
select the "best“ tree-scale
wavelet packet decomposition
problem : reducing the amount of data needed to represent e fingerprint image in 7.76(a).
Using three-scale wavelet packet trees, there are 87,522 potential decompositions.
Figure 7.76(b) shows one of them.
One reasonable criterion for selecting a decomposition for the compression the image
of 7.76(a) is the energy content ,include the dimensional function
( J.Shanbehzadeh M.Gholizadeh )
For each node of the analysis tree, beginning with the root and proceeding
level by level to the leaves:
Step 1. Compute both the energy of the node, denoted E (for parent
energy), and the energy of its four offspring.
Step 2. If the combined energy of the offspring is less than the energy of the
Parent include the offspring in the analysis tree. If the combined energy
of the offspring is greater than or equal to that of the parent, prune the
offspring, keeping only the parent. It is a leaf of the optimized analysis tree.
( J.Shanbehzadeh M.Gholizadeh )
many of the original full packet decomposition's 64 subbands in Fig. 7.76(b) have been
eliminated. In addition, the subimages that are not split (further decomposed) in Fig. 7.77 are relatively smooth and composed of pixels
that are middle gray in value.
( J.Shanbehzadeh M.Gholizadeh )