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Sampling theorems

Let us consider finite energy signals f ¯ 2 = | f ¯ ( x ) | 2 d x of finite support, which is normalized to [ 0 , 1 ] or [ 0 , 1 ] 2 for images. A sampling process implementsa filtering of f ¯ ( x ) with a low-pass impulse response φ ¯ s ( x ) and a uniform sampling to output a discrete signal:

f [ n ] = f ¯ φ ¯ s ( n s ) for 0 n < N .

In two dimensions, n = ( n 1 , n 2 ) and x = ( x 1 , x 2 ) . These filtered samples can also be written as inner products:

f ¯ φ ¯ s ( n s ) = f ( u ) φ ¯ s ( n s - u ) d u = f ( x ) , φ s ( x - n s )

with φ s ( x ) = φ ¯ s ( - x ) . Chapter 3 explains that φ s is chosen, like in the classic Shannon–Whittaker sampling theorem, so that a family of functions { φ s ( x - n s ) } 1 n N is a basis of an appropriate approximation space U N . The best linear approximation of f ¯ in U N recovered from these samples is the orthogonal projection  f ¯ N of f in U N , and if the basis is orthonormal, then

f ¯ N ( x ) = n = 0 N - 1 f [ n ] φ s ( x - n s ) .

A sampling theorem states that if f ¯ U N then f ¯ = f ¯ N so recovers f ¯ ( x ) from the measured samples. Most often, f ¯ does not belong to this approximation space. It is called aliasing in the context of Shannon–Whittaker sampling, where U N is the space of functions having a frequency support restricted to the N lower frequencies. The approximation error f ¯ - f ¯ N 2 must then be controlled.

Linear approximation error

The approximation error is computed by finding an orthogonal basis B = { g ¯ m ( x ) } 0 m < + of the whole analog signal space L 2 ( R ) [ 0 , 1 ] 2 , with the first N vector { g ¯ m ( x ) } 0 m < N that defines an orthogonal basis of U N . Thus, the orthogonal projection on U N can be rewritten as

f ¯ N ( x ) = m = 0 N - 1 f ¯ , g ¯ m g ¯ m ( x ) .

Since f ¯ = m = 0 + f ¯ , g ¯ m g ¯ m , the approximation error is the energy of the removed inner products:

ϵ l ( N , f ) = f ¯ - f ¯ N 2 = m = N + | f ¯ , g ¯ m | 2 .

This error decreases quickly when N increases if the coefficient amplitudes | f ¯ , g ¯ m | have a fast decay when the index m increases. The dimension N is adjusted to the desired approximation error.

Figure (a) shows a discrete image f [ n ] approximated with N = 256 2 pixels. Figure(c) displays a lower-resolution image f N / 16 projected on a space U N / 16 of dimension N / 16 , generated by N / 16 large-scale wavelets. It is calculated by setting all thewavelet coefficients to zero at the first two smaller scales. The approximation error is f - f N / 16 2 / f 2 = 14 × 10 - 3 . Reducing the resolution introduces more blur and errors.A linear approximation space U N corresponds to a uniform grid that approximates precisely uniform regular signals.Since images f ¯ are often not uniformly regular, it is necessary to measure it at a high-resolution N . This is why digital cameras have a resolution that increases as technology improves.

Sparse nonlinear approximations

Linear approximations reduce the space dimensionality but can introduce important errors when reducing the resolutionif the signal is not uniformly regular, as shown by Figure(c). To improve such approximations, more coefficients should be kept where needed—notin regular regions but near sharp transitions and edges.This requires defining an irregular sampling adapted to the local signal regularity. This optimized irregular sampling has a simple equivalent solutionthrough nonlinear approximations in wavelet bases.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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ramon Reply
Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
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Azam
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, A wavelet tour of signal processing, the sparse way. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10711/1.3
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