Image transforms

IMAGE TRANSFORMS.

MAHESH MOHAN.M.RGECT S3 ECEROLL NO: 5

OM NAMA SIVAYA

SHIVA KUDE KANANE…

DISCRETE COSINE TRANSFORM

1

0

1

0

),(2

)12(cos

2

)12(cos

)()(2),(

M

i

N

j

jifN

vj

M

ui

MN

vCuCvuF

Given a function f(i, j) over two integer variables i and j, the 2D DCT transforms it into a new function F(u, v), with integers u and v running over the same range as i and j such that

where i,u = 0, ..., M – 1 and j,v = 0, ..., N – 1 and

otherwise

xifxC

1

02

1

)(

SIGNIFICANCE OF DCT

The entries in Y will be organized based on the human visual system.

The most important values to our eyes will be placed in the upper left corner of the matrix.

The least important values will be mostly in the lower right corner of the matrix.

Horizontal freq

Most

Important

Ver

tical

fre

q

Semi-

Important

Least

Important

DCT MATRIX

DEMONSTRATION OF DCTCan You Tell the Difference?

ORIGINAL Base layer (MSE =38.806)

DCT MATRIX

PROPERTIES OF DCT

1.Decorrelation

Normalized autocorrelation of uncorrelated image before and after DCT

Normalized autocorrelation of correlated image before and after DCT

PROPERTIES OF DCT

2. Energy Compaction

GECT and its DCT

PROPERTIES OF DCT

3. Seperability

A serious drawback in transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place.

DRAWBACK OF DCT

HISTORY OF WAVELET

1805 Fourier analysis developed1965 Fast Fourier Transform (FFT) algorithm1980’s beginnings of wavelets in physics, vision, speech processing 1986 Mallat unified the above work1985 Morlet & Grossman continuous wavelet transform …asking: how can you get perfect reconstruction without redundancy?1985 Meyer tried to prove that no orthogonal wavelet other than Haar exists, found one by trial and error!1987 Mallat developed multiresolution theory, DWT, wavelet construction techniques (but still noncompact)1988 Daubechies added theory: found compact, orthogonal wavelets with arbitrary number of vanishing moments!

1990’s: wavelets took off, attracting both theoreticians and engineers

• For many applications, you want to analyze a function in both space and frequency

• Analogous to a musical score

WHY WAVELET TRANSFORM

Discrete transforms give you frequency information, smearing space.Samples of a function give you temporal information, smearing frequency.

These basis functions or baby wavelets are obtained from a single prototype wavelet called the mother wavelet, by dilations or contractions (scaling) and translations (shifts).

WAVELET BASIS

WAVELET BASIS (contd)

The wavelets are generated from a single basic wavelet , the so-called mother wavelet, by scaling and translation.

s

t

sts

1

)(,

DISCRETE WAVELET TRANSFORM

· Discrete wavelet is written as

j

j

jkj s

skt

st

0

00

0

,

1)(

j and k are integers and s0 > 1 is a fixed scaling step. The translation factor t0 depends on the scaling step. The effect of discretizing the wavelet is that the time-scale space is now sampled at discrete intervals.

0

1)()( *

,, dttt nmkj If j=m and k=n

others

DISCRETE WAVELET TRANSFORMFILTER bANK APPROXIMATION.

But wind up with twice as much data as we started with. To correct this problem, downsampling is introduced.

DISCRETE WAVELET TRANSFORMFILTER bANK APPROXIMATION.

· The original signal, S, passes through two complementary filtersand emerges as two signals .

PRACTICAL EXAMPLE OF FILTER bANK APPROXIMATION.

RECONSTRUCTION OF FILTER bANK APPROXIMATION.

RECONSTRUCTION OF FILTER bANK APPROXIMATION.

Wavelet Decomposition

Multiple-Level Decomposition

The decomposition process can be iterated, so that one signal is broken down into many lower-resolution components. This is called the wavelet decomposition tree.

2d DWT

Shiva kathone,,,,kude kannane

1 level Haar 2 level HaarOriginal

NEED FOR A NEW TRANSFORM?

Efficiency of a representation refers to the ability to capture significant information about an object of interest using a small description.

Wavelet Curvelet

WHAT WE WISH in ATRANSFORM?

Multiresolution. The representation should allow images to be successively approximated, from coarse to fine resolutions.

Localization. The basis elements in the representation should be localized in both the spatial and the frequency domains.

Critical sampling. For some applications (e.g., compression), the representation should form a basis, or a frame with small redundancy.

Directionality. The representation should contain basis elements oriented at a variety of directions

Anisotropy. To capture smooth contours in images, the representation should contain basis elements using a variety of elongated shapes with different aspect ratios.

CONTOURLET TRANSFORM

• Captures smooth contours and edges at any orientation

• Filters noise.• Derived directly from discrete domain

instead of extending from continuous domain.

• Can be implemented using filter banks.

CONTOURLET TRANSFORM

The transform decouples the multiscale and the directional decompositions.

4

DEMONSTRATION CONTOURLET TRANSFORM

01

1

2

2

33 4

5

5

6

6

7

7 8

8

16

9

9

10

10

11

11

12

12

13

13

14

1415

15

16

10

0 12

34

5

6

7 8

15 161314

11 12

9 5

4

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