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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

SNR 10 * log 10 u ¯ - E ( u ¯ ) 2 u ¯ - u 2 ,

where u ¯ is the original image and E ( u ¯ ) is the mean intensity value of 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 β increases because the approximation problems become closer to the original ones. However, the alternating algorithmsconverge slowly when β is large, which is well-known for the class of penalty methods. An effective remedy is to graduallyincrease β 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 .

Continuation vs. no continuation: u * is an“exact”solution corresponding to β = 2 14 . The horizontal axis represents the number of iterations, and thevertical axis is the relative error e k = u k - u * / u * .

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 β . As can be seen from , with continuation on β the convergence is greatly sped up. In our experiments, we implemented the alternating minimizationalgorithms with continuation on β , which we call the resulting algorithm“Fast Total Variation de-convolution”or FTVd, which, for TV/L 2 , the framework is given below.


  • Input f , K and μ > 0 . Given β max > β 0 > 0 .
  • Initialize u = f , u p = 0 , β = β 0 and ϵ > 0 .
  • While β β max , Do
    • Run Algorithm "Basic Algorithm" until an optimality condition is met.
    • β 2 * β .
  • End Do
SNRs of images recovered from () for different β .
Results recovered from TV/L 2 . Image Man is blurred by a Gaussian kernel, while image Lena is blurred by across-channel kernel. Gaussian noise with zero mean and standard deviation 10 - 3 is added to both blurred images. The left images are the blurry and noisy observations, and the right ones arerecovered by FTVd.

Generally, it is difficult to determine how large β 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 β reached a reasonably large value. To see this, we plot the SNR values of restored images corresponding to β = 2 0 , 2 1 , , 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 β 2 7 . This suggests that β need not to be excessively large from a practical point of view. In our experiments, we set β 0 = 1 and β max = 2 7 in Algorithm  "Practical Implementation" . For each β , the inner iteration was stopped once an optimality condition is satisfied. For TV/L 1 problems, we also implement continuation on γ , 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 γ = 2 15 and β = 2 10 in the approximation model and implemented continuation on both β and γ .

Results recovered from TV/L 1 . Image Lena is blurred by a cross-channel kernel and corrupted by 40 % (left) and 50 % (right) random-valued noise. The top row contains the blurry and noisy observations and the bottom row shows the resultsrecovered by FTVd.

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.


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

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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