Wavelets (Chapter 7) CS474/674 – Prof. Bebis. STFT (revisited) Time/Frequency localization depends on window size. Once you choose a particular window

Wavelets (Chapter 7)

CS474/674 – Prof. Bebis

STFT (revisited)

• Time/Frequency localization depends on window size.• Once you choose a particular window size, it will be the same

for all frequencies.• Many signals require a more flexible approach - vary the

window size to determine more accurately either time or frequency.

The Wavelet Transform

• Overcomes the preset resolution problem of the STFT by using a variable length window:

– Narrower windows are more appropriate at high frequencies (better time localization)

– Wider windows are more appropriate at low frequencies (better frequency localization)

The Wavelet Transform (cont’d)

Wide windows do not provide good localization at high frequencies.


A narrower window is more appropriate at high frequencies.


Narrow windows do not provide good localization at low frequencies.


A wider window is more appropriate at low frequencies.

What is a wavelet?

• It is a function that “waves” above and below the x-axis; it has (1) varying frequency, (2) limited duration, and (3) an average value of zero.

• This is in contrast to sinusoids, used by FT, which have infinite energy.

Sinusoid Wavelet

Wavelets

• Like sine and cosine functions in FT, wavelets can define basis functions ψk(t):

• Span of ψk(t): vector space S containing all functions f(t) that can be represented by ψk(t).

( ) ( )k kk

f t a t

Wavelets (cont’d)

• There are many different wavelets:

MorletHaar Daubechies

(dyadic/octave grid)

Basis Construction - Mother Wavelet

( ) jk t

Basis Construction - Mother Wavelet (cont’d)

scale =1/2j

(1/frequency)

/2( ) 2 2 j jjk t t k

j

k

Continuous Wavelet Transform (CWT)

1( , )

t

tC s f t dt

ss

translation parameter, measure of time

scale parameter (measure of frequency)

Mother wavelet (window)normalization

constant

ForwardCWT:

scale =1/2j=1/frequency

CWT: Main Steps

1. Take a wavelet and compare it to a section at the start of the original signal.

2. Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. The higher C is, the more the similarity.

CWT: Main Steps (cont’d)

3. Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal.

CWT: Main Steps (cont’d)

4. Scale the wavelet and repeat steps 1 through 3.

5. Repeat steps 1 through 4 for all scales.

Coefficients of CTW Transform

1( , )

t

tC s f t dt

ss

• Wavelet analysis produces a time-scale view of the input signal or image.

Continuous Wavelet Transform (cont’d)

• Inverse CWT:

1( ) ( , ) ( )

s

tf t C s d ds

ss

note the double integral!

FT vs WT

weighted by F(u)

weighted by C(τ,s)

Properties of Wavelets

• Simultaneous localization in time and scale- The location of the wavelet allows to explicitly represent

the location of events in time.

- The shape of the wavelet allows to represent different detail or resolution.

Properties of Wavelets (cont’d)

• Sparsity: for functions typically found in practice, many of the coefficients in a wavelet representation are either zero or very small.

• Adaptability: Can represent functions with discontinuities or corners more efficiently.

• Linear-time complexity: many wavelet transformations can be accomplished in O(N) time.

1( ) ( , ) ( )

s

tf t C s d ds

ss

Discrete Wavelet Transform (DWT)

( ) ( )jk jkk j

f t a t

/2( ) 2 2 j jjk t t k

(inverse DWT)

(forward DWT)

where

*( ) ( )jkjkt

a f t t

DFT vs DWT

• FT expansion:

• WT expansion

or

one parameter basis

( ) ( )l ll

f t a t

( ) ( )jk jkk j

f t a t

two parameter basis

Multiresolution Representation using

( ) ( )jk jkk j

f t a t

( )f t

( )jk t

j

finedetails

coarsedetails

wider, large translations


( ) ( )jk jkk j

f t a t

( )f t

( )jk t

j

finedetails

coarsedetails


( ) ( )jk jkk j

f t a t

( )f t

( )jk t

j

finedetails

coarsedetails

narrower, small translations


high resolution

(more details)

low resolution

(less details)

…

( ) ( )jk jkk j

f t a t

( )f t

1ˆ ( )f t

2ˆ ( )f t

ˆ ( )sf t

( )jk t

j

Prediction Residual Pyramid - Revisited

• In the absence of quantization errors, the approximation pyramid can be reconstructed from the prediction residual pyramid.

• Prediction residual pyramid can be represented more efficiently.

(with sub-sampling)

Efficient Representation Using “Details”

details D2

L0

details D3

details D1

(without sub-sampling)

Efficient Representation Using Details (cont’d)

representation: L0 D1 D2 D3

A wavelet representation of a function consists of (1)a coarse overall approximation (2)detail coefficients that influence the function at various scales.

in general: L0 D1 D2 D3…DJ

Reconstruction (synthesis)H3=H2+D3

details D2

L0

details D3

H2=H1+D2

H1=L0+D1

details D1

(without sub-sampling)

Example - Haar Wavelets

• Suppose we are given a 1D "image" with a resolution of 4 pixels:

[9 7 3 5]

• The Haar wavelet transform is the following:

L0 D1 D2 D3(with sub-sampling)

Example - Haar Wavelets (cont’d)

• Start by averaging the pixels together (pairwise) to get a new lower resolution image:

• To recover the original four pixels from the two averaged pixels, store some detail coefficients.

1

[9 7 3 5]


• Repeating this process on the averages (i.e., low resolution image) gives the full decomposition:

1

Harr decomposition:


• We can reconstruct the original image by adding or subtracting the detail coefficients from the lower-resolution versions.

2 1 -1


Detail coefficientsbecome smaller andsmaller as j increases.

Dj

Dj-1

D1L0

How tocompute Di ?

Multiresolution Conditions

• If a set of functions can be represented by a weighted sum of ψ(2jt - k), then a larger setlarger set, including the original, can be represented by a weighted sum of ψ(2j+1t - k):

low resolution

high resolution

j

Multiresolution Conditions (cont’d)

• If a set of functions can be represented by a weighted sum of ψ(2jt - k), then a larger set, including the original, can be represented by a weighted sum of ψ(2j+1t - k):

Vj: span of ψ(2jt - k): ( ) ( )j k jkk

f t a t

Vj+1: span of ψ(2j+1t - k): 1 ( 1)( ) ( )j k j kk

f t b t

1j jV V

Nested Spaces Vj

ψ(t - k)

ψ(2t - k)

ψ(2jt - k)

…

V0

V1

Vj

Vj : space spanned by ψ(2jt - k)

Multiresolution conditions nested spanned spaces:

( ) ( )jk jkk j

f t a t f(t) ϵ Vj

Basis functions:

i.e., if f(t) ϵ V j then f(t) ϵ V j+1

1j jV V

How to compute Di ? (cont’d)

( ) ( )jk jkk j

f t a t f(t) ϵ Vj

IDEA: define a set of basisfunctions that span thedifference between Vj+1 and Vj

Orthogonal Complement Wj

• Let Wj be the orthogonal complement of Vj in Vj+1

Vj+1 = Vj + Wj


If f(t) ϵ Vj+1, then f(t) can be represented using basis functions φ(t) from Vj+1:

1( ) (2 )jk

k

f t c t k

( ) (2 ) (2 )j jk jk

k k

f t c t k d t k Vj+1 = Vj + Wj

Alternatively, f(t) can be represented using two sets of basis functions, φ(t) from Vj and ψ(t) from Wj:

Vj+1

Think of Wj as a means to represent the parts of a function in Vj+1 that cannot be represented in Vj

1( ) (2 )jk

k

f t c t k

( ) (2 ) (2 )j jk jk

k k

f t c t k d t k Vj Wj


differencesbetweenVj and Vj+1


• using recursion on Vj:

( ) ( ) (2 )jk jk

k k j

f t c t k d t k

V0 W0, W1, W2, …basis functions basis functions

Vj+1 = Vj-1+Wj-1+Wj = …= V0 + W0 + W1 + W2 + … + Wj

if f(t) ϵ Vj+1 , then:

Vj+1 = Vj + Wj

Summary: wavelet expansion (Section 7.2)

• Wavelet decompositions involve a pair of waveforms (mother wavelets):

φ(t) ψ(t)encode lowresolution info

encode details orhigh resolution info

( ) ( ) (2 )jk jk

k k j

f t c t k d t k

scaling function wavelet function

1D Haar Wavelets

• Haar scaling and wavelet functions:

computes average computes details

φ(t) ψ(t)

1D Haar Wavelets (cont’d)

• V0 represents the space of one pixel images

• Think of a one-pixel image as a function that is constant over [0,1)

Example:0 1

width: 1


• V1 represents the space of all two-pixel images

• Think of a two-pixel image as a function having 21 equal-sized constant pieces over the interval [0, 1).

• Note that

Examples: 0 ½ 1

0 1V V

= +

width: 1/2

e.g.,


• V j represents all the 2j-pixel images

• Functions having 2j equal-sized constant pieces over interval [0,1).

• Note that

Examples: width: 1/2j

ϵ Vj ϵ Vj

1j jV V

e.g.,


V0, V1, ..., V j are nested

i.e.,

VJ

…

V1

V0coarse details

fine details

1j jV V

Define a basis for Vj

• Mother scaling function:

• Let’s define a basis for V j :

0 1

note alternative notation: ( ) ( )ji jix x

1

Define a basis for Vj (cont’d)

Define a basis for Wj (cont’d)

• Mother wavelet function:

• Let’s define a basis ψ ji for Wj :

1-1

0 1/2 1

( ) ( )ji jix x note alternative notation:

Define a basis for Wj

Notethat the dotproductbetween basisfunctions in Vj

and Wj is zero!


Basis functions ψ ji of W j

Basis functions φ ji of V j

form a basis in V j+1


V3 = V2 + W2


V2 = V1 + W1


V1 = V0 + W0

Example - Revisited

f(x)=

V2

φ2,0(x)

φ2,1(x)

φ2,2(x)

φ2,3(x)

Example (cont’d)

V2

f(x)=

Example (cont’d)

V1and W1

V2=V1+W1

φ1,0(x)

φ1,1(x)

ψ1,0(x)

ψ1,1(x)

(divide by 2 for normalization)

Example (cont’d)

Example (cont’d)

V2=V1+W1=V0+W0+W1

V0 ,W0 and W1

φ0,0(x)

ψ0,0(x)

ψ1,0 (x)

ψ1,1(x)

(divide by 2 for normalization)

Example

Example (cont’d)

Filter banks

• The lower resolution coefficients can be calculated from the higher resolution coefficients by a tree-structured algorithm (i.e., filter bank).

h0(-n) is a lowpass filter and h1(-n) is a highpass filter

Subbandencoding(analysis)

Example (revisited)

[9 7 3 5]low-pass,down-sampling

high-pass, down-sampling

(9+7)/2 (3+5)/2 (9-7)/2 (3-5)/2 V1 basis functions

Filter banks (cont’d)

Next level:

Example (revisited)

[9 7 3 5]

high-pass, down-sampling

low-pass,down-sampling

(8+4)/2 (8-4)/2

V1 basis functions

Convention for illustrating 1D Haar wavelet decomposition

x x x x x x … x x

detail

average

…

re-arrange:

re-arrange:

V1 basis functions

Examples of lowpass/highpass (analysis) filters

Daubechies

Haarh0

h1

h0

h1


• The higher resolution coefficients can be calculated from

the lower resolution coefficients using a similar structure.

Subbandencoding(synthesis)


Next level:

Examples of lowpass/highpass (synthesis) filters

Daubechies

Haar (same as for analysis):

+

g0

g1

g0

g1

2D Haar Wavelet Transform

• The 2D Haar wavelet decomposition can be computed using 1D Haar wavelet decompositions.– i.e., 2D Haar wavelet basis is separable

• Two decompositions (i.e., correspond to different basis functions):– Standard decomposition

– Non-standard decomposition

Standard Haar wavelet decomposition

• Steps

(1) Compute 1D Haar wavelet decomposition of each row of the original pixel values.

(2) Compute 1D Haar wavelet decomposition of each column of the row-transformed pixels.

Standard Haar wavelet decomposition (cont’d)

x x x … xx x x … x

… … .x x x ... x

(1) row-wise Haar decomposition:

…

detail

average

…

… … .

…

…

… … .

re-arrange terms


(1) row-wise Haar decomposition:

…

detail

average

…

…… … .

…

… … . …

row-transformed result


(2) column-wise Haar decomposition:

…

detail

average

…

…… … .

…

…

…… … .

…

row-transformed result column-transformed result

Example

…

…

…… … .

row-transformed result

…

… … .

re-arrange terms

Example (cont’d)

…

…

…… … .

column-transformed result

2D Haar basis for standard decomposition

To construct the standard 2D Haar wavelet basis, consider all possible outer products of 1D basis functions.

φ0,0(x)

ψ0,0(x)

ψ1,0(x)

ψ1,1(x)

V2=V0+W0+W1

Example:

2D Haar basis for standard decomposition

To construct the standard 2D Haar wavelet basis, consider all possible outer products of 1D basis functions.

φ00(x), φ00(x) ψ00(x), φ00(x) ψ10(x), φ00(x)

( ) ( )ji jix x ( ) ( )j

i jix x

2D Haar basis of standard decomposition

( ) ( )ji jix x

( ) ( )ji jix x

V2

Non-standard Haar wavelet decomposition

• Alternates between operations on rows and columns.

(1) Perform one level decomposition in each row (i.e., one step of horizontal pairwise averaging and differencing).

(2) Perform one level decomposition in each column from step 1 (i.e., one step of vertical pairwise averaging and differencing).

(3) Repeat the process on the quadrant containing averages only (i.e., in both directions).

Non-standard Haar wavelet decomposition (cont’d)

x x x … xx x x … x

… … .x x x . . . x

one level, horizontalHaar decomposition:

…

…

… … .

…

…

…

… … .

one level, vertical Haar decomposition:

…

Non-standard Haar wavelet decomposition (cont’d)

one level, horizontalHaar decompositionon “green” quadrant

one level, vertical Haar decompositionon “green” quadrant

……

… … .

……

re-arrange terms

…

…

…… … .

…

Example

……

… … .

……

re-arrange terms

Example (cont’d)

…

…

…… … .

2D Haar basis for non-standard decomposition

• Defined through 2D scaling and wavelet functions:

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

x y x y

x y x y

x y x y

x y x y

000 ( , ) ( , )

( , ) 2 (2 ,2 )

( , ) 2 (2 ,2 )

( , ) 2 (2 ,2 )

j j j jkf

j j j jkf

j j j jkf

x y x y

x y x k y f

x y x k y f

x y x k y f

2D Haar basis for non-standard decomposition (cont’d)

( ) ( )ji jix x

( ) ( )ji jix x

V2

2D Haar basis for non-standard decomposition (cont’d)

000 ( , ) ( , )

( , ) 2 (2 ,2 )

( , ) 2 (2 ,2 )

( , ) 2 (2 ,2 )

j j j jkf

j j j jkf

j j j jkf

x y x y

x y x k y f

x y x k y f

x y x k y f

LL

LH: intensity variations along columns (horizontal edges)

HL: intensity variations along rows (vertical edges)

HH: intensity variations along diagonals

LL: average Detail coefficients

• Three sets of detail coefficients (i.e., subband encoding)

2D DWT using filter banks (analysis)

LL LH

HL HH


LLH LHV HLD HH

The wavelet transform is applied again on the LL sub-image

LL LLLH

HL HH


2D Inverse DWT using filter banks (synthesis)

H LHV HLD HH

Wavelets Applications

• Noise filtering

• Image compression– Fingerprint compression

• Image fusion

• RecognitionG. Bebis, A. Gyaourova, S. Singh, and I. Pavlidis, "Face Recognition by Fusing Thermal Infrared and Visible Imagery", Image and Vision Computing, vol. 24, no. 7, pp. 727-742, 2006.

• Image matching and retrieval Charles E. Jacobs Adam Finkelstein David H. Salesin, "Fast

Multiresolution Image Querying", SIGRAPH, 1995.

Fast Multiresolution Image Querying

painted low resolution target

queries

Charles E. Jacobs Adam Finkelstein David H. Salesin, "Fast Multiresolution Image Querying", SIGRAPH, 1995.

Documents

Wavelets (Chapter 7) CS474/674 – Prof. Bebis. STFT (revisited) Time/Frequency localization depends on window size. Once you choose a particular window