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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

find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Please were did you get y=mx+b from
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
consider r(a+b) = ra + rb. The a and b are the trig identity.
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
i do not understand anything
lol...it gets better
I've been struggling so much through all of this. my final is in four weeks 😭
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
thank you I have heard of him. I should check him out.
is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
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