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Chapter 7(7.1,7.2,7.3)

Wavelets and Multiresolution Processing

7.1 Preliminaries

When looking at images, we generally see connected regions of similar texture and intensity

levels combined to form objects. Small or low contrast objects are better viewed at high

resolution. If small and large objects are present it can be advantageous to study them at

different resolutions. From math viewpoint, images are 2D arrays of intensity values with

locally varying statistics that result from different features.

Image pyramids

An image pyramid is a collection of decreasing resolution images arranged in the shape of a

pyramid. The base of a pyramid is a high resolution image being processed; the apex contains a

low-resolution approximation. While moving up, both size and resolution decrease.

The apex level 0 is of size 1x1. Most pyramids are truncated toP+ 1 levels, where

1PJ. The total number of pixels in aP+ 1 level pyramid is


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On the diagram for constructing two image pyramids, the level j-1 approximation output

provides the images needed to build an approximation pyramid, while the level jprediction

residual output is used to build a complementary prediction residual pyramid. Unlikeapproximation pyramids, prediction residual pyramids contain only one reduced-resolution

approximation of the input image (top of the pyramid, level J-P). All other levels contain

prediction residuals where the level j prediction residual (J-P+1 j J) is defined as the

difference between level j approximation and its estimate. An estimate of the level j

approximation is computed based on the level j-1 approximation. Approximation and

prediction residual pyramids can be computed by iterations. The original image is placed in

levelJof the approximation pyramid. The pyramids then are generated inPiterations for the

following steps forj = J, J-1,, J-P+1:

1. Compute a reduced-resolution approximation of the level j image by filtering and

downsampling the filtered image by 2.

2. Estimate the level j input image from the reduced-resolution approximation by upsamplingby and filtering.

3. Compute the difference between the output of step 2 and the input of step 1. place this result

in levelj of the prediction residual pyramid.


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Both operations are usually followed by approximation and interpolation filters.

4-level approximation pyramid: an original 512x512 image and its approximations at 256x256,

128x128, and 64x64. A Gaussian smoothing filter was used. Prediction residual pyramid A

bilinear interpolation filter was used.


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Subband coding

Filterh0(n) is a low-pass (half-band) filter, whose outputflp(n) is an approximation of the input

f(n); h1(n) is a high-pass (half-band) filter, whose outputfhp(n) is a high-frequency ordetailpart

of the inputf(n). Synthesis filtersglp(n) andghp(n) combine two subband signals to produce f

(n) The goal of subband coding is to select filters such that Which is called perfect

reconstruction conditions that requiref ^(n)=

f(n)

g0(n) = (1)n h 1(n)

g1(n)= (1)n+1h 0(n)

or

g0(n) = (1)n+1h 1 (n)

g1(n)= (1)nh 0 (n)

Filters must satisfy biorthogonality condition

Of special interest are filters satisfying orthonormality condition

which satisfy the conditions


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An orthonormal filter bank can be designed from a singleprototype filter; all other filters are

computed from the prototype.


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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 matrix that 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.

Consider a signal f in one dimension from - to +

Haar scaling function is denoted by (t) and Haar wavelet function is denoted by (t). Haar scaling function (averaging or lowpass filter) at level 0 (in the original signal) is

given by

Translation by j is denoted by j (x)

Figure below shows both (x) and j (x)


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7.2 Multiresolution Expansions

In Multi-resolution Analysis (MRA), a Scaling Function is used to create a series of

approximations of a function or image, each differing by a factor 2 from its nearest neighboringapproximations. Additional functions, called Wavelet, are used to encode the difference in

information between adjacent approximations.

Series expansions

If is an orthonormal basis forV, then

( ) ( )k kk

f x x = : real-valued expansion coefficients( ) : real-valued expansion functions

k

k x

*( ), ( ) ( ) ( ) ( ): the dual function of ( )k k k k k x f x x f x dx x x = = % % %

( ) ( )k k

x x = %


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If f(x) is an element of V0, it is also an element of V1

all V0 expansion functions are a part of V1

V0 is a subspace


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7.3 Wavelet Transforms in One Dimension


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Discrete Wavelet Transform

Cwt is redundant as the transform is calculated by continuously shifting a continuouslyscalable function over a signal and calculating the correlation between the two

The discrete form is normally a [piecewise] continuous function obtained by samplingthe time-scale space at discrete intervals.

The process of transforming a continuous signal into a series of wavelet coefficients isknown as wavelet series decomposition.

Scaling function can be expressed in wavelets from - to j. Adding a wavelet spectrum to the scaling function yields a new scaling function, with

a spectrum twice as wide as the firsto Addition allows us to express the first scaling function in terms of the second.

o The formal expression of this phenomenon leads to multiresolution formulation or two-scale

relation as

o This equation states that the scaling function (average) at a given scale can be expressed in

terms of translated scaling functions at the next smaller scale, where the smaller scale implies

more detail


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