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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.

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