# 9.1 A class of fast algorithms for total variation image restoration  (Page 2/6)

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Deblurring or decovolution aims to recover the unknown image $\overline{u}\left(x\right)$ from $f\left(x\right)$ and $k\left(x\right)$ based on ( ). When $k\left(x\right)$ is unknown or only an estimate of it is available, recovering $\overline{u}\left(x\right)$ from $f\left(x\right)$ is called blind deconvolution. Throughout this module, we assume that $k\left(x\right)$ is known and $\omega \left(x\right)$ is either Gaussian or impulsive noise. When $k\left(x\right)$ is equal to the Dirac delta, the recovery of $\overline{u}\left(x\right)$ becomes a pure denoising problem. In the rest of this section, we review theTV-based variational models for image restoration and introduce necessary notation for analysis.

## Total variation for image restoration

The TV regularization was first proposed by Rudin, Osher and Fatemi in for image denoising, and then extended to image deblurring in . The TV of $u$ is defined as

$\begin{array}{c}\hfill \mathrm{TV}\left(u\right)={\int }_{\Omega }\parallel \nabla u\left(x\right)\parallel \phantom{\rule{0.277778em}{0ex}}\mathrm{d}x.\end{array}$

When $\nabla u\left(x\right)$ does not exist, the TV is defined using a dual formulation , which is equivalent to ( ) when $u$ is differentiable. We point out that, in practical computation, discrete forms of regularization are always used wheredifferential operators are replaced by ceratin finite difference operators.We refer TV regularization and its variants as TV-like regularization. In comparison toTikhonov-like regularization, the homogeneous penalty on image smoothness in TV-like regularization can better preserve sharp edgesand object boundaries that are usually the most important features to recover. Variational modelswith TV regularization and ${\ell }_{2}$ fidelity has beenwidely studied in image restoration; see e.g. , and references therein. For ${\ell }_{1}$ fidelity with TV regularization, itsgeometric properties are analyzed in , , . The superiority of TV over Tikhonov-like regularization was analyzedin , for recovering images containing piecewise smooth objects.

Besides Tikhonov and TV-like regularization, there are other well studied regularizers in the literature, e.g. the Mumford-Shahregularization . In this module, we concentrate on TV-like regularization. We derive fast algorithms, study theirconvergence, and examine their performance.

## Discretization and notation

As used before, we let $\parallel ·\parallel$ be the 2-norm. In practice, we always discretize an image defined on $\Omega$ , and vectorize the two-dimensional digitalized image into a long one-dimensional vector. We assume that $\Omega$ is a square region in ${\mathbb{R}}^{2}$ . Specifically, we first discretize $u\left(x\right)$ into a digital image represented by a matrix $U\in {\mathbb{R}}^{n×n}$ . Then we vectorize $U$ column by column into a vector $u\in {\mathbb{R}}^{{n}^{2}}$ , i.e.

$\begin{array}{c}\hfill {u}_{i}={U}_{pq},\phantom{\rule{1.em}{0ex}}i=1,...,{n}^{2},\end{array}$

where ${u}_{i}$ denotes the $i$ th component of $u$ , ${U}_{pq}$ is the component of $U$ at $p$ th row and $q$ th column, and $p$ and $q$ are determined by $i=\left(q-1\right)n+p$ and $1\le q\le n$ . Other quantities such as the convolution kernel $k\left(x\right)$ , additive noise $\omega \left(x\right)$ , and the observation $f\left(x\right)$ are all discretized correspondingly. Now we present the discrete forms of the previously presented equations.The discrete form of ( ) is

$\begin{array}{c}\hfill f=K\overline{u}+\omega ,\end{array}$

where in this case, $\overline{u},\omega ,f\in {\mathbb{R}}^{{n}^{2}}$ are all vectors representing, respectively, the discrete forms of the originalimage, additive noise and the blurry and noisy observation, and $K\in {\mathbb{R}}^{{n}^{2}×{n}^{2}}$ is a convolution matrix representing the kernel $k\left(x\right)$ . The gradient $\nabla u\left(x\right)$ is replaced by certain first-order finite difference at pixel $i$ . Let ${D}_{i}\in {\mathbb{R}}^{2×{n}^{2}}$ be a first-order local finite difference matrix at pixel $i$ in horizontal and vertical directions. E.g. when the forward finite difference is used, we have

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
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
oops. ignore that.
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Commplementary angles
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Sherica
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Sherica
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Tamia
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
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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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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
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Damian
silver nanoparticles could handle the job?
Damian
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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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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