De-noising on the Body Centered Cubic (BCC) Sampling Lattice Tai - - PowerPoint PPT Presentation

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De-noising on the Body Centered Cubic (BCC) Sampling Lattice Tai - - PowerPoint PPT Presentation

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De-noising on the Body Centered Cubic (BCC) Sampling Lattice Tai Meng CMPT775 2006/Spring 4/13/2006 1 Motivation Why is BCC superior to Cartesian (CC) in medical imaging? 3D: saves 30% samples 3D time-varying: saves 50%

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4/13/2006 1

De-noising on the Body Centered Cubic (BCC) Sampling Lattice

Tai Meng CMPT775 – 2006/Spring

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2 4/13/2006

Motivation

Why is BCC superior to Cartesian (CC) in

medical imaging?

3D: saves 30% samples 3D time-varying: saves 50% samples Higher dimensions: potentially higher savings

BCC grid seems well-positioned to take

  • ver the CC grid in medical imaging
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Motivation

Why is BCC not used in medical imaging?

Few tools for the BCC grid exist

Why de-noising on the BCC lattice?

De-noising is necessary in medical imaging De-noising tools for BCC does not exist

Ideal goal: BCC is no worse than CC in

de-noising

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Abstract

Two types of noise investigated

Salt & pepper noise: impulse noise Gaussian white noise: random noise

Two filters investigated

Median filter: salt & pepper noise Gaussian smoothing filter: white noise

Error plots of CC vs BCC de-noising

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The BCC Lattice

Start with canonical

CC lattice

A lattice point belongs

to the BCC lattice if and only if all three of its coordinates are even, or if all three are odd

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Sampling Equivalence

Theorem

Consider an unknown 3D signal. On average,

to capture the same amount of information via sampling, it takes the BCC lattice roughly 70%

  • f the number of samples that it would take

the CC lattice

In the limit, the exact percentage is 1/sqrt(2)

~= 70.7%

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Stage 1: Sampler

Marschner Lobb (ML) dataset CC: 64 x 64 x64 samples BCC: 45 x 45 x 90 samples The number of samples in BCC dataset is

roughly 70% of that of CC dataset

By theorem, they capture roughly the

same amount of information from ML

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Stage 1: Sampler

CC/BCC pair: noise free CC/BCC pair: salt & pepper noise

Input: probability of noise

CC/BCC pair: Gaussian noise

Input: standard deviation = average difference

  • f noise samples
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Salt & Pepper Noise Generation

Input: probability p For each sample, generate y = rand[0..1]

If 0 <= y < p/2, set sample to 0 (pepper) If p/2 <= y <= p, set sample to 254 (salt) Else leave sample alone

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White Noise Generation

Method 1: The Central Limit Theorem

states that the sum of N random numbers will approach normal distribution as N approaches infinity; N >= 30 works

Method 2: Rejection sampling; dart

throwing till a sample falls under the Gaussian envelope; very slow

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White Noise Generation

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Stage 1: De-noiser

Input: filter radius, noise type, grid type Noise type, grid type -> choose filter

Set that filter to the input filter radius Apply the filter to the dataset

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Stage 1: De-noiser

Four filters to choose from:

CC median filter BCC median filter CC Gaussian filter BCC Gaussian filter

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Salt & Pepper: Low Noise

CC Original CC Salt & Pepper 3% Filter size = 1.414214 Neighborhood size = 19 BCC Original BCC Salt & Pepper 3% Filter size = 1.415730 Neighborhood size = 15

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Salt & Pepper: Medium Noise

CC Original CC Salt & Pepper 10% Filter size = 1.414214 Neighborhood size = 19 BCC Original BCC Salt & Pepper 10% Filter size = 1.415730 Neighborhood size = 15

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Salt & Pepper: High Noise

CC Original CC Salt & Pepper 20% Filter size = 1.414214 Neighborhood size = 19 BCC Original BCC Salt & Pepper 20% Filter size = 1.415730 Neighborhood size = 15

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Salt & Pepper: High Noise

CC Original CC Salt & Pepper 20% Filter size = 1 Neighborhood size = 7 BCC Original BCC Salt & Pepper 20% Filter size = 1.226058 Neighborhood size = 9

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White Noise: Low Noise

CC Original CC Gaussian Sigma = 2 Filter size = 2.236068 Neighborhood size = 57 BCC Original BCC Gaussian Sigma = 2 Filter size = 2.347723 Neighborhood size = 51

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White Noise: Medium Noise

CC Original CC Gaussian Sigma = 7 Filter size = 2.236068 Neighborhood size = 57 BCC Original BCC Gaussian Sigma = 7 Filter size = 2.347723 Neighborhood size = 51

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White Noise: High Noise

CC Original CC Gaussian Sigma = 14 Filter size = 2.236068 Neighborhood size = 57 BCC Original BCC Gaussian Sigma = 14 Filter size = 2.347723 Neighborhood size = 51

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Stage 2: Data Plot

Error metric after de-noise:

Compute difference between de-noised

dataset and noise-free dataset

Mean ~= 0, so can be ignored Use standard deviation as error metric

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Stage 2: Data Plot

One 2D point for each neighbourhood

Filter radius corresponding to neighbourhood Error after filtering with this radius

Generate 10 points for first 10

neighbourhoods

Can already see a convergent behaviour Larger neighbourhood => slower filtering

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Plot: Low Noise

Index CC Radius CC Size BCC Radius BCC Size 1 2 3 4 5 6 7 8 9 10 0.000000 1 1.000000 7 1.414214 19 1.732051 27 2.000000 33 2.236068 57 2.449490 81 2.828427 93 3.000000 123 3.162278 147 0.000000 1 1.226058 9 1.415730 15 2.002145 27 2.347723 51 2.452117 59 2.831461 65 3.085513 89 3.165669 113 3.467817 137

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Plot: Medium Noise

Index CC Radius CC Size BCC Radius BCC Size 1 2 3 4 5 6 7 8 9 10 0.000000 1 1.000000 7 1.414214 19 1.732051 27 2.000000 33 2.236068 57 2.449490 81 2.828427 93 3.000000 123 3.162278 147 0.000000 1 1.226058 9 1.415730 15 2.002145 27 2.347723 51 2.452117 59 2.831461 65 3.085513 89 3.165669 113 3.467817 137

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Plot: High Noise

Index CC Radius CC Size BCC Radius BCC Size 1 2 3 4 5 6 7 8 9 10 0.000000 1 1.000000 7 1.414214 19 1.732051 27 2.000000 33 2.236068 57 2.449490 81 2.828427 93 3.000000 123 3.162278 147 0.000000 1 1.226058 9 1.415730 15 2.002145 27 2.347723 51 2.452117 59 2.831461 65 3.085513 89 3.165669 113 3.467817 137

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Conclusion

Median filtering

BCC seems comparable to CC

Gaussian filtering

BCC seems better for low noise levels CC seems better for higher noise levels Need further investigation

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Bonus: Neighborhood Plot

Investigate the claim that the ratio of BCC

  • ver CC neighbourhood sizes converge to

roughly 0.7 (i.e. 1/sqrt(2))

Plotted first 50 BCC/CC ratios

Can see convergent behaviour Know that in the limit, ratio = 1/sqrt(2)

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Bonus: Neighborhood Plot

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References

Robert Fisher, Simon Perkins, Ashley

Walker and Erik Wolfart. Digital Filters. Department of Artificial Intelligence, University of Edinburgh, UK, 2003.

Complete list:

http://www.taimeng.com/grad_school/CMPT7

75_proj/references.htm

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Thank You!