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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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