A Gentle Introduction to Bilateral Filtering and its Applications
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Transcript of A Gentle Introduction to Bilateral Filtering and its Applications
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A Gentle Introductionto Bilateral Filteringand its Applications
A Gentle Introductionto Bilateral Filteringand its Applications
07/10: Novel Variantsof the Bilateral Filter
Jack Tumblin – EECS, Northwestern University
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Review: Bilateral FilterReview: Bilateral Filter
A 2-D filter window: weights vary with intensityA 2-D filter window: weights vary with intensity
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DomainDomain
RangeRange
f(x)f(x)xx
2 Gaussian Weights:2 Gaussian Weights:product = product = ellisoidal footprintellisoidal footprint
Normalize weights toNormalize weights toalways sum to 1.0always sum to 1.0
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Review: Bilateral FilterReview: Bilateral Filter
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Why it works: graceful segmentationWhy it works: graceful segmentation• Smoothing for ‘similar’ parts Smoothing for ‘similar’ parts ONLYONLY• Range Gaussian Range Gaussian ss acts as a ‘filtered region’ finder acts as a ‘filtered region’ finder DomainDomain
RangeRange
f(x)f(x)xx
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Bilateral Filter VariantsBilateral Filter Variants
• before the ‘Bilateral’ name :
– Yaroslavsky (1985): T.D.R.I.M.
– Smith & Brady (1997): SUSAN
And now, a growing set of named variants:
• ‘Trilateral’ Filter (Choudhury et al., EGSR 2003)
• Cross-Bilateral (Petschnigg04, Eisemann04)
• NL-Means (Buades 05)
And more coming: application driven…
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Who was first? Many PioneersWho was first? Many Pioneers
• Elegant, Simple, Broad Idea
‘Invented’ several times
• Different Approaches, Increasing Clarity
– Tomasi & Manduchi(1998): ‘Bilateral Filter’
– Smith & Brady (1995): ‘SUSAN’ “Smallest Univalue Segment Assimilating Nucleus”
– Yaroslavsky(1985) ‘Transform Domain Image Restoration Methods’
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New Idea!New Idea!1985 Yaroslavsky: 1985 Yaroslavsky:
A 2-D filter window: A 2-D filter window: weights vary with intensity ONLY weights vary with intensity ONLY
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DomainDomain
RangeRange
f(x)f(x)xx
Square neighborhood,Square neighborhood,Gaussian WeightedGaussian Weighted‘‘similarity’similarity’
Normalize weights toNormalize weights toalways sum to 1.0always sum to 1.0
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New Idea!New Idea!1995 Smith: ‘SUSAN’ Filter1995 Smith: ‘SUSAN’ Filter
A 2-D filter window: weights vary with intensityA 2-D filter window: weights vary with intensity
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DomainDomain
RangeRange
f(x)f(x)xx
2 Gaussian Weights:2 Gaussian Weights:product = product = ellisoidal footprintellisoidal footprint
Normalize weights toNormalize weights toalways sum to 1.0always sum to 1.0
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Background:Background: ‘Unilateral’ Filter ‘Unilateral’ Filter
e.g. traditional, linear, FIR filterse.g. traditional, linear, FIR filtersKey Idea:Key Idea: ConvolutionConvolution
- Output(x) = local weighted avg. of inputs.- Output(x) = local weighted avg. of inputs.- Weights vary within a ‘window’ of nearby x- Weights vary within a ‘window’ of nearby x
• Smoothes away details, Smoothes away details, BUTBUT blurs result blurs result
ccweight(x)weight(x)
Note that weightsNote that weightsalways sum to 1.0always sum to 1.0
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Piecewise smooth result Piecewise smooth result – averages local small details, ignores outliersaverages local small details, ignores outliers– preserves steps, large-scale ramps, and curves,...preserves steps, large-scale ramps, and curves,...
• Equivalent to anisotropic diffusion and robust statisticsEquivalent to anisotropic diffusion and robust statistics [Black98,Elad02,Durand02][Black98,Elad02,Durand02]
• Simple & Fast Simple & Fast (esp. w/ (esp. w/ [Durand02][Durand02] FFT-based speedup) FFT-based speedup)
Bilateral Filter:Bilateral Filter: Strengths Strengths
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Output at is Output at is
average of a average of a tiny regiontiny region
Bilateral Filter:Bilateral Filter: 3 Difficulties 3 Difficulties
• Poor Smoothing in Poor Smoothing in High Gradient RegionsHigh Gradient Regions
• Smoothes and bluntsSmoothes and bluntscliffs, valleys & ridgescliffs, valleys & ridges
• Can combine disjoint Can combine disjoint signal regions signal regions
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Bilateral Filter:Bilateral Filter: 3 Difficulties 3 Difficulties
• Poor Smoothing in Poor Smoothing in High Gradient RegionsHigh Gradient Regions
• Smoothes and bluntsSmoothes and bluntscliffs, valleys & ridgescliffs, valleys & ridges
• Can combine disjoint Can combine disjoint signal regions signal regions
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‘‘Blunted Corners’ Blunted Corners’ Weak Halos Weak Halos
Bilateral :Bilateral :
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‘‘Blunted Corners’ Blunted Corners’ Weak Halos Weak Halos
‘‘Trilateral’:Trilateral’:
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Bilateral Filter:Bilateral Filter: 3 Difficulties 3 Difficulties
• Poor Smoothing in Poor Smoothing in High Gradient RegionsHigh Gradient Regions
• Smoothes and bluntsSmoothes and bluntscliffs, valleys & ridgescliffs, valleys & ridges
• Disjoint regions Disjoint regions can blend together can blend together
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New Idea!New Idea!Trilateral Filter (Choudhury 2003)Trilateral Filter (Choudhury 2003)
Goal:Goal: Piecewise linear smoothing, not piecewise constantPiecewise linear smoothing, not piecewise constant
Method:Method:Extensions to the Bilateral FilterExtensions to the Bilateral Filter
PositionPosition
IntensityIntensity
EXAMPLE:EXAMPLE: remove noise from a piecewise linear scanline remove noise from a piecewise linear scanline
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Outline: BilateralOutline: BilateralTrilateral FilterTrilateral Filter
Three Key Ideas:Three Key Ideas:• TiltTilt the filter window the filter window
according to bilaterally-according to bilaterally-smoothed gradientssmoothed gradients
• LimitLimit the filter window the filter window to connected regions to connected regions of similar smoothed gradient.of similar smoothed gradient.
• AdjustAdjust Parameters Parameters from measurements from measurements of the windowed signalof the windowed signal
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Outline: BilateralOutline: BilateralTrilateral FilterTrilateral Filter
Key Ideas:Key Ideas:• TiltTilt the filter window the filter window
according to bilaterally-according to bilaterally-smoothed gradientssmoothed gradients
• LimitLimit the filter window the filter window to connected regions to connected regions of similar smoothed gradient.of similar smoothed gradient.
• AdjustAdjust Parameters Parameters from measurements from measurements of the windowed signalof the windowed signal
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Outline: BilateralOutline: BilateralTrilateral FilterTrilateral Filter
Key Ideas:Key Ideas:• TiltTilt the filter window the filter window
according to bilaterally-according to bilaterally-smoothed gradientssmoothed gradients
• LimitLimit the filter window the filter window to connected regions to connected regions of similar smoothed gradient.of similar smoothed gradient.
• AdjustAdjust Parameters Parameters from measurements from measurements of the windowed signalof the windowed signal
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Comparisons: Skylight DetailsComparisons: Skylight Details
..
BilateralBilateral
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Comparisons: Skylight DetailsComparisons: Skylight Details
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TrilateralTrilateral
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. .
• ,,
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Trilateral Filter (Choudhury 2003)Trilateral Filter (Choudhury 2003)
• StrengthsStrengths– Sharpens Sharpens cornerscorners– Smoothes similar Smoothes similar gradientsgradients– Automatic Automatic parameter parameter settingsetting– 3-D 3-D mesh de-noisingmesh de-noising, too!, too!
• WeaknessesWeaknesses– S-L-O-W;S-L-O-W; very costly connected-region finder very costly connected-region finder– Shares Bilateral’s Shares Bilateral’s ‘Single-pixel region’ artifacts‘Single-pixel region’ artifacts– Noise ToleranceNoise Tolerance limits; disrupts ‘tilt’ estimates limits; disrupts ‘tilt’ estimates
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NEW IDEA : ‘Joint’ or ‘Cross’ Bilateral’ Petschnigg(2004) and Eisemann(2004)NEW IDEA : ‘Joint’ or ‘Cross’ Bilateral’ Petschnigg(2004) and Eisemann(2004)
Bilateral two kinds of weights
NEW : get them from two kinds of images.
• Smooth image A pixels locally, but
• Limit to ‘similar regions’ of image B
Why do this? To get ‘best of both images’
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Ordinary Bilateral FilterOrdinary Bilateral Filter
Bilateral two kinds of weights, one image A :
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Image A:
DomainDomain
RangeRange
f(x)f(x)xx
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‘Joint’ or ‘Cross’ Bilateral Filter‘Joint’ or ‘Cross’ Bilateral Filter
NEW: two kinds of weights, two images
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A: Noisy, dim(ambient image)
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B: Clean,strong (Flash image)
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Image A: Warm, shadows, but too Noisy(too dim for a good quick photo)(too dim for a good quick photo)
Image A: Warm, shadows, but too Noisy(too dim for a good quick photo)(too dim for a good quick photo)
No-flash
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Image B: Cold, Shadow-free, Clean(flash: simple light, ALMOST no shadows)(flash: simple light, ALMOST no shadows)
Image B: Cold, Shadow-free, Clean(flash: simple light, ALMOST no shadows)(flash: simple light, ALMOST no shadows)
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MERGE BEST OF BOTH: apply‘Cross Bilateral’ or ‘Joint Bilateral’
MERGE BEST OF BOTH: apply‘Cross Bilateral’ or ‘Joint Bilateral’
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(it really is much better!)(it really is much better!)
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Recovers Weak Signals Hidden by NoiseRecovers Weak Signals Hidden by Noise
Noisy but Strong…
Noisy and Weak…
+ Noise =
+ Noise =
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Ordinary Bilateral Filter? Ordinary Bilateral Filter?
Noisy but Strong…
Noisy and Weak…
BF
BF
Step feature GONE!!Step feature GONE!!
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Ordinary Bilateral Ordinary Bilateral
Noisy but Strong…
Noisy and Weak…
Signal too small to rejectSignal too small to reject
Range filter preserves signalRange filter preserves signal
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‘Cross’ or ‘Joint’ Bilateral Idea:‘Cross’ or ‘Joint’ Bilateral Idea:
Noisy but Strong…
Noisy and Weak…
Range filter preserves signalRange filter preserves signal
Use stronger signal’s range Use stronger signal’s range filter weights…filter weights…
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‘Joint’ or ‘Cross’ Bilateral FilterPetschnigg(2004) and Eisemann(2004)
‘Joint’ or ‘Cross’ Bilateral FilterPetschnigg(2004) and Eisemann(2004)
• Useful Residues. To transfer details,
– CBF(A,B) to remove A’s noisy details
– CBF(B,A) to remove B’s clean details;
– add to CBF(A,B) – clean, detailed image!
•CBF(A,B): smoothes image A only;(e.g. no flash)
•Limits smoothing to stay within regions where Image B is ~uniform (e.g. flash)
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New Idea:NL-Means Filter (Buades 2005)
New Idea:NL-Means Filter (Buades 2005)
• Same goals: ‘Smooth within Similar Regions’
• KEY INSIGHT: Generalize, extend ‘Similarity’
– Bilateral: Averages neighbors with similar intensities;
– NL-Means: Averages neighbors with similar neighborhoods!
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
• For each and
every pixel p:
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
• For each and
every pixel p:
– Define a small, simple fixed size neighborhood;
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
• For each and
every pixel p:
– Define a small, simple fixed size neighborhood;
– Define vector Vp: a list of neighboring pixel values.
Vp = 0.740.320.410.55………
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
‘Similar’ pixels p, q
SMALL vector distance;
|| Vp – Vq ||2
p
q
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
‘Dissimilar’ pixels p, q
LARGE vector distance;
|| Vp – Vq ||2
pq
q
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
‘Dissimilar’ pixels p, q
LARGE vector distance;
Filter with this!
|| Vp – Vq ||2
pq
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
p, q neighbors define
a vector distance;
Filter with this:No spatial term!
|| Vp – Vq ||2 pq
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p qp 2||||||||1
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NL-Means Method:Buades (2005)NL-Means Method:Buades (2005)
pixels p, q neighborsSet a vector distance;
Vector Distance to p sets weight for each pixel q
|| Vp – Vq ||2 pq
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p2||||
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NL-Means Filter (Buades 2005)NL-Means Filter (Buades 2005)
• Noisysourceimage:
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NL-Means Filter (Buades 2005)NL-Means Filter (Buades 2005)
• GaussianFilter
Low noise,
Low detail
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NL-Means Filter (Buades 2005)NL-Means Filter (Buades 2005)
• AnisotropicDiffusion
(Note ‘stairsteps’:~ piecewiseconstant)
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NL-Means Filter (Buades 2005)NL-Means Filter (Buades 2005)
• BilateralFilter
(better, butsimilar‘stairsteps’:
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NL-Means Filter (Buades 2005)NL-Means Filter (Buades 2005)
• NL-Means:
Sharp,
Low noise,
Few artifacts.
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Many More Possibilities: EXPERIMENT!Many More Possibilities: EXPERIMENT!
• Bilateral goals are subjective;
‘Local smoothing within similar regions’‘Edge-preserving smoothing’‘Separate large structure & fine detail’‘Eliminate outliers’‘Filter within edges, not across them’
• It’s simplicity invites new inventive answers.
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