3. 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
Verticalfreq
Semi-
Important
Least
Important
DCT MATRIX
8. 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
9. HISTORY OF WAVELET
1805 Fourier analysis developed
1965 Fast Fourier Transform (FFT) algorithm
1980’s beginnings of wavelets in physics, vision, speech processing
1986 Mallat unified the above work
1985 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
10. • 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.
11. 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
12. WAVELET BASIS (contd)
The wavelets are generated from a single basic wavelet , the so-
called mother wavelet, by scaling and translation.
s
t
s
ts
1
)(,
13. DISCRETE WAVELET TRANSFORM
Discrete wavelet is written as
j
j
j
kj
s
skt
s
t
0
00
0
,
1
)(
j and k are integers and s0 > 1 is a fixed scaling step. The translation factor 0
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
15. But wind up with twice as much data as we started with. To
correct this problem, downsampling is introduced.
DISCRETE WAVELET TRANSFORM
FILTER bANK APPROXIMATION.
The original signal, S, passes through two complementary filters
and emerges as two signals .
22. 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
23. 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.
24. 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.