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y = x + w = z u + w .

The BLS-GSM algorithm is as follows:

  • Decompose the image into subbands
  • For the HH, HL, and LH subbands:
    • Compute the noise covariance, C w , from the image-domain noise covariance
    • Estimate C y , the noisy neighborhood covariance
    • Estimate C u using C u = C y + C w
    • Compute Λ and M , where Q , Λ is the eigenvector/eigenvalue expansion of the matrix S - 1 C u S - T S is the symmetric square root of the positive definite matrix C w , and M = SQ
    • For each neighborhood
      • For each value z in the integration range
        • Compute E [ x c | y , z ] = n = 1 N z m c n λ n v n z λ n + 1 , where m i j M , v v = M - 1 y , λ = d i a g ( λ ) , and c is the index of the reference coefficient.
        • Compute the conditional density p ( y | z )
      • Compute the posterior p ( z | y )
      • Compute E [ x c | y ]
    Reconstruct the denoised image from the processed subbands and the lowpass residual

Denoising simulation

Simulation description

In order to compare and evaluate the efficacies of the Bishrink and BLS-GSM algorithms for the purpose of denoising image data, a simulation was developed to quantitatively examine their performance after addition of random noise to otherwise approximately noiseless images with a variety of features representative of those found in astronomical images. Specifically, the images encoded in the widely available files Moon.tif, which primarily demonstrates smoothly curving attributes, and Cameraman.tif, which exhibits a range of both smooth and coarse features, distributed in the MATLAB image processing toolbox were considered.

As a preliminary preparation for the simulation, the images were preprocessed such that they were represented in the form of a grayscale pixel matrix taking values on the interval [ 0 , 1 ] of square dimensions equal to a convenient power of two. Noisy versions of each image were generated by superposition of a random matrix with Gaussian distributed pixel elements on the image matrix, using noise variance values { . 01 , . 1 , 1 } . For each noise variance level and original image, 100 contaminated images were created in this way using a set of 100 different random generator seeds, which was the same for each noise level and original image. A redundant discrete wavelet transform of each of these contaminated images was computed using the length 8 Daubechies filters, and the denoised wavelet coefficients were estimated using both the Bishrink and the BLS-GSM algorithms as previously described. Computation of the inverse redundant discrete wavelet transform using the denoised wavelet coefficients then yielded 100 images denoised with the Bishrink algorithm and 100 images denoised with the BLS-GSM algorithm for each original image and noise variance level.

Using this simulated data, the performance of the two denoising methods on each image at each noise contamination level were evaluated using the six statistical measures described here. The first of these was the mean square error M S E , which is calculated by the average of

1 n i = 1 n f x i - f ^ x i 2

over all 100 denoisings. Related to the above was the root mean square error R M S E , which is calculated by computing the square root of the mean square error. A third was the root mean square bias R M S B , which is calculated by

1 n i = 1 n f x i - f ¯ x i 2

where f ¯ x i is the average of f ^ x i over all 100 denoisings. Two more, the maximum deviation M X D V , calculated by the average of

max 1 < i < n f x i - f ^ x i

over all 100 denoisings, and L1, calculated by the average of

i = 1 n f x i - f ^ x i

over all 100 denoisings, were also examined. The results of this simulation now follow.

Bishrink results

Simulation measures for noise variance .01
Measure Cameraman Moon
MSE 0.0019 0.0004
RMSE 0.0442 0.0188
L1 2019.9 3160.4
RMSB 0.0274 0.0117
MXDV 0.3309 0.2634
Cameraman with noise variance .01
Moon with noise variance .01
Simulation measures for noise variance .1
Measure Cameraman Moon
MSE 0.0063 0.0012
RMSE 0.0296 0.0345
L1 3612.4 5880.7
RMSB 0.0568 0.0213
MXDV 0.6147 0.4116
Cameraman with noise variance .1
Moon with noise variance .1
Simulation measures for noise variance 1
Measure Cameraman Moon
MSE 0.0173 0.0052
RMSE 0.1315 0.0722
L1 6183.7 11839
RMSB 0.0934 0.0389
MXDV 0.8991 0.9774
Cameraman with noise variance 1
Moon with noise variance 1

Bls-gsm results

Simulation measures for noise variance .01
Measure Cameraman Moon
MSE 0.0015 0.0003
RMSE 0.0390 0.0165
L1 1711.0 2718.6
RMSB 0.0283 0.0141
MXDV 0.3192 0.2635
Cameraman with noise variance .01
Moon with noise variance .01
Simulation measures for noise variance .1
Measure Cameraman Moon
MSE 0.0052 0.0008
RMSE 0.0718 0.0288
L1 3111.5 4786.5
RMSB 0.0583 0.0224
MXDV 0.5862 0.3337
Cameraman with noise variance .1
Moon with noise variance .1
Simulation measures for noise variance 1
Measure Cameraman Moon
MSE 0.0136 0.0017
RMSE 0.1167 0.0410
L1 5283.5 1500.2
RMSB 0.0970 0.0346
MXDV 0.7750 0.4614
Cameraman with noise variance 1
Moon with noise variance 1

Conclusions

The results obtained from this simulation now allow us to evaluate and comment upon the suitability of each of the two methods examined for the analysis of astronomical image data. As is clearly manifested in the quantitative simulation results, the BLS-GSM algorithm demonstrated more accurate performance than did the Bishrink algorithm in every measure consistently over all pictures and noise levels. That does not, however, indicate that it would be the method of choice in all circumstances. While BLS-GSM outperformed the Bishrink algorithm in the denoising simulation, the measures calculated for the Bishrink algorithm indicate that it also produced a reasonably accurate image estimate. Also, the denoised images produced by the Bishrink simulation exhibit a lesser degree of qualitative smoothing of fine features like the craters of the moon and grass of the field. The smoothing observed with the BLS-GSM algorithm could make classification of fine, dim objects difficult as they are blended into the background. Thus, the success of the Bishrink algorithm in preserving fine signal details while computing an accurate image estimate is likely to outweigh overall accuracy in applications searching for small, faint objects such as extrasolar planets, while the overall accuracy of the BLS-GSM algorithm recommend it for coarse and bright featured images.

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
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Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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China
Cied
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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