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

 Page 6 / 6

We tested several kinds of blurring kernels including Gaussian, average and motion. The additive noise is Gaussian for TV/L ${}^{2}$ problems and impulsive for TV/L ${}^{1}$ problem. The quality of image is measured by the signal-to-noise ratio (SNR) defined by

$\text{SNR}\triangleq 10*{log}_{10}\frac{\parallel \overline{u}-E\left(\overline{u}\right){\parallel }^{2}}{\parallel \overline{u}{-u\parallel }^{2}},$

where $\overline{u}$ is the original image and $E\left(\overline{u}\right)$ is the mean intensity value of $\overline{u}$ . All blurring effects were generated using the MATLAB function“imfilter " with periodic boundaryconditions, and noise was added using“imnoise ". All theexperiments were finished under Windows Vista Premium and MATLAB v7.6 (R2008a) running on a Lenovo laptop with an Intel Core 2 DuoCPU at 2 GHz and 2 GB of memory.

## Practical implementation

Generally, the quality of the restored image is expected to increase as $\beta$ increases because the approximation problems become closer to the original ones. However, the alternating algorithmsconverge slowly when $\beta$ is large, which is well-known for the class of penalty methods. An effective remedy is to graduallyincrease $\beta$ from a small value to a pre-specified one. compares the different convergence behaviors of the proposed algorithm when with and without continuation, where weused Gaussian blur of size 11 and standard deviation 5 and added white Gaussian noise with mean zero and standard deviation ${10}^{-3}$ .

In this continuation framework, we compute a solution of an approximation problem which used a smaller beta, and use thesolution to warm-start the next approximation problem corresponding to a bigger $\beta$ . As can be seen from , with continuation on $\beta$ the convergence is greatly sped up. In our experiments, we implemented the alternating minimizationalgorithms with continuation on $\beta$ , which we call the resulting algorithm“Fast Total Variation de-convolution”or FTVd, which, for TV/L ${}^{2}$ , the framework is given below.

[FTVd]:

• Input $f$ , $K$ and $\mu >0$ . Given ${\beta }_{max}>{\beta }_{0}>0$ .
• Initialize $u=f$ , ${u}_{p}=0$ , $\beta ={\beta }_{0}$ and $ϵ>0$ .
• While $\beta \le {\beta }_{max}$ , Do
• Run Algorithm "Basic Algorithm" until an optimality condition is met.
• $\beta ←2*\beta$ .
• End Do

Generally, it is difficult to determine how large $\beta$ is sufficient to generate a solution that is close to be a solution ofthe original problems. In practice, we observed that the SNR values of recovered images from the approximation problems are stabilizedonce $\beta$ reached a reasonably large value. To see this, we plot the SNR values of restored images corresponding to $\beta ={2}^{0},{2}^{1},\cdots ,{2}^{18}$ in . In this experiment, we used the same blur and noise as we used in the testing ofcontinuation. As can be seen from , the SNR values on both images essentially remain constant for $\beta \ge {2}^{7}$ . This suggests that $\beta$ need not to be excessively large from a practical point of view. In our experiments, we set ${\beta }_{0}=1$ and ${\beta }_{max}={2}^{7}$ in Algorithm  "Practical Implementation" . For each $\beta$ , the inner iteration was stopped once an optimality condition is satisfied. For TV/L ${}^{1}$ problems, we also implement continuation on $\gamma$ , and used similar settings as used in TV/L ${}^{2}$ .

## Recovered results

In this subsection, we present results recovered from TV/L ${}^{2}$ and TV/L ${}^{1}$ problems including ( ), ( ) and their multichannel extensions. We tested various of blurs with differentlevels of Gaussian noise and impulsive noise. Here we merely present serval test results. gives two examples of blurry and noisy images and the recovered ones, where the blurredimages are corrupted by Gaussian noise, while gives the recovered results where the blurred images are corrupted by random-valued noise. For TV/L ${}^{1}$ problems, we set $\gamma ={2}^{15}$ and $\beta ={2}^{10}$ in the approximation model and implemented continuation on both $\beta$ and $\gamma$ .

## Concluding remarks

We proposed, analyzed and tested an alternating algorithm FTVd which for solving the TV/ ${L}^{2}$ problem. This algorithm was extended to solve the TV/ ${L}^{1}$ model and their multichannel extensions by incorporating an extension of TV. Cross-channel blurs are permittedwhen the underlying image has more than one channels. We established strong convergence results for the algorithms and validated a continuationscheme. Numerical results are given to demonstrate the feasibility and efficiency of the proposed algorithms.

## Acknowledgements

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundation grant DMS-0739420.

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
what is hormones?
Wellington
Got questions? Join the online conversation and get instant answers!