Aviva: Public

ECE462 – Lecture 16

• Matrix based calculation of wavelet transform

Given any Nx1 vector X = x[0],……….,x[N-1] we would like to find a matrix

A : NXN so that A XT will be equivalent to one-level wavelet transform

For example for N = 4 the Haar WT matrix is

A1 =

1

2

1

20 0

0 01

2

1

21

2−

1

20 0

0 01

2−

1

2

Or if we normalize the filters so that energy is preserved

Or if we normalize the filters so that energy is preserved

A =

1

2

1

20 0

0 01

2

1

21

2−

1

20 0

0 01

2−

1

2

Notice before that A1 A1T = 1

4I

A AT = I = AT A

That is A1 is orthogonal while

A is orthogonal matrix

For applications in images and by considering separable filters one level wavelet decomposition works as follows:

Given the image F(i,j), NxN

Obtain A . F . AT

Notice that in order to get back from the WT to the image we should proceed as follows

AT(A F AT) A = F

For a multiple level WT we must define matrices A1: NxN , A2 : N

2xN

2, A3 :

N

4xN

4and so on

and apply these consequently to the LL sub image of each level

Example: Consider our previous example for the 4x4 image

F:

1 0 0 01 0 0 01 0 0 01 1 1 1

• First level WT : (Haar)

A =

1

2

1

20 0

0 01

2

1

21

2−

1

20 0

0 01

2−

1

2

1

2x

2 0 2 03 2 1 00 0 0 0−1 −2 1 1

• Second level WT on F1 : 1

2

2 03 2

Now

A2 =

1

2

1

21

2−

1

2

1

4

7 3−3 1

So the overall 2 – level W.T

7

4

3

41 0

−3

4

1

4

1

20

0 0 0 0

−1

2−1

1

20

≡

To get back to original image we proceed as follows

First:

A1T .

𝐿𝐿2 𝐿𝐻2𝐻𝐿2 𝐻𝐻2

A1 = F1

2x2 2x2 2x2 2x2

Then

A1 T 𝐿𝐿1 𝐿𝐻1𝐻𝐿1 𝐻𝐻1

A1 F

4x4 4x4 4x4 4x4 (to be assigned as an exercise)

LL2

• Wavelet based image compression and distortions

The sub-images at different resolutions are processed by a spectrum estimator for bit allocations. Statistical properties of the sub-images guide this quantization step.

➢The lowpass sub-band (upper left) is allocated the most bits

➢ Strategy based on the rate distortion curve:

Given B bits, how do we allocate bk bits per pixel to the k-th sub-image so that the reconstructed image has smallest distortion?

• Distortion measures: MSE, PSNR, ….

At medium bit rate (0.25 bpp) (that is about 32:1) the objective measures are good indicators of the subjective quality of the image

At low bit rates this is not always the case

• Possible artifacts in reconstructed images

Blurring border distortions, blocking, checker boarding, ringing depend on

1) Filter choices

2) Bit allocation

3) Convolutions

Blurring occurs when we do not assign enough bits to the higher sub-bands

In biorthogonal filter decomposition analysis and synthesis filters are different

Lowpass synthesis: Long and smooth to avoid blocking and checker boarding

Hipass synthesis: Short to avoid ringing

The non-smooth scaling function Checker board effects