11 Wavelets Handouts

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Image AnalysisWavelets and Multiresolution Processing

Niclas Brlin [email protected]

Department of Computing ScienceUme University

February 20, 2009

Niclas Brlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 1 / 24

Time vs. frequency resolution

Each coefficient of a Fourier spectra of a signal (image) providesinformation about the signal contents of that frequency.

However, the Fourier coefficients contain no spatial knowledge.

On the other hand, each spatial coefficient, i.e. sample, contains

no frequency information.

In image analysis, different frequencies correspond to objects ofdifferent sizes.

Low frequencies correspond to slow changes over a large area.

High frequenciescorrespond to fast changes over a small area.

Waveletsallow us to analyse a combination of spatial and

frequency information.


The Fourier vs. Wavelet transformation

The Fourier transform uses sinusoids of infinite durationas basis

functions.

Wavelet transforms uses small waves (wavelets) of limited

duration as basis functions.

The Fourier transform gives information about images frequency

decomposition.The Wavelet transforms have resolution in the frequency domain

as well as in the spatial domain (what frequencies are in the

image and where).

Wavelets are conceptually similar to musical scores: Which tones, and when to play them?


The Image PyramidA closely related concept is the Image pyramid.

Images of different spatial resolutions, form levelsof the pyramid.

The original image with the highest resolution is at level J, the

lowest pyramid level.

Each higher level contain a lower resolution image, usually half.

The image at the top Level 0 contain only one pixel.

The total number of pixels in a P+ 1 level pyramid is

N2

1 +1

(4)1+

1

(4)2+ + 1

(4)P

4

3N2.



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The Approximation and Prediction Residual PyramidsLevel j of the Approximation Pyramid is formed by smoothing

and downsampling the image at level j + 1.

Level j of the Prediction Residual Pyramidis formed as the

difference between the upsampled and interpolatedlevel j 1image, and the level j approximation image.


The Approximation and Prediction Residual Pyramids


Subband coding and filter banks

The concept of subband coding is to split a signal into two (ormore) components that completely describes the input.

This is performed in conjunction with two filter banks.

The analysis filter bank uses filters h0(n) and h1(n) to split theinput sequence f(n) into two half-length sequences (subbands)flp(n) and fhp(n).

The synthesis filter bankg0(n) and g1(n) combines flp(n) andfhp(n) to form the reconstructed signal f(n).


Perfect reconstruction

The lowpass filter h0(n) produce the approximation subbandflp(n).

The highpass filter h1(n) produce the detail subbandfhp(n).

If the filter banks are chosen properly, the signal can be perfectly

reconstructed from its subbands flp(n) and fhp(n).

If the analysis filter bank is recursively applied to the

approximation subband, we obtain an approximation pyramid.



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2D subbandsSubband coding for a 2D signal is separable and may be applied

sequentially along the rows and columns.

Furthermore, the downsampling can be applied once after each

analysis stage, significantly improving performance.

The result is four subbands: The approximation a(m, n), thevertical detaildV(m, n), the horizontal detaildH(m,n), and thediagonal detaildD(m, n).


2D subband example


Multiresolution analysis

Multiresolution analysis (MRA) uses a scaling function(x) tocreate a series of approximations of a signal or image, each

differing by a factor of 2 from its nearest neighboring

approximation.

Wavelet functions(x) are then used to encode the difference(detail) in information between adjacent approximations.

MRA defines a set of requirements for the scaling functions.

Given a scaling function that meets these requirements we

define a wavelet function to use.


The Haar Transform

The Haar transform basis functions are the oldest and simplest

known orthonormal wavelets.

The Haar transform is separable and expressible in matrix form

T = HFHT,

where F is an N N image, H is an N N transformation matrixthat contains the Haar basis functions, and T is the resulting

N N transform.The basis functions are scaled and translated versions of a

mother wavelet.



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The Haar TransformAssume N = 2n, 0 p n 1, k = 2p + q 1, and

q =

0 or 1, p = 0

1 q 2p, p= 0 .

Then the Haar basis functionshk(z) are defined as

h0(z) = h00(z) =1N, z [0, 1], and

hk(z) = hpq(z) =1N

2p/2 (q 1)/2p z< (q 0.5)/2p,2p/2 (q 0.5)/2p


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Time-frequency resolution

Each tile represent an equal portion of the time-frequency plane.

There is a trade-off between the resolution in time and frequency.

Higher resolution in frequency (low frequencies) lowerresolution in time.

Higher resolution in time (high frequencies) lower resolution infrequency.


The Discrete 2D wavelet transform


The Fast Wavelet Transform (FWT)

The Fast Wavelet Transform (FWT) is essentially a recursive

interleaving application of the smoothing ((x)) and differencing((x)) filters and the downsampling.

The first application corresponds to the highest frequencies(narrowest wavelets).

Each successive downsampling corresponds to widening of the

wavelet by a factor of two.


2D discrete wavelet transform using Haar basis



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Multiscale DWT


Wavelet edge detection


Wavelet noise removal


Other wavelets

The optimal wavelet function is called Daubechies and have

self-similar, i.e. fractal, characteristics.


11 Wavelets Handouts

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