Image Filtering - Vision Labsvision.psych.umn.edu/.../Papers/Lec07ImageFiltering.pdf ·...
Transcript of Image Filtering - Vision Labsvision.psych.umn.edu/.../Papers/Lec07ImageFiltering.pdf ·...
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
Image Filtering
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
Topics
• Filtering as a linear transform
• Fourier Transform
• Filtering as feature finding
• Filtering for target detection
• Edge detection
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
€
For a filter size N by M,Rij = HuvFi−u, j−v
v=1:M∑
u=1:N∑
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Convolution• Represent these weights as an
image, H
• H is usually called the kernel
• Operation is called convolution• Properties:
• Convolution is commutative.
• Convolution is associative.
• Convolution is distributive.
• Result is:
• Notice the order ofindices– all examples can be put
in this form
– it’s a result of thederivation expressing anyshift-invariant linearoperator as aconvolution.
€
Rij = Hi−u, j−vFuvu,v∑
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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Example: Smoothing by Averaging
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Smoothing with a Gaussian• Smoothing with an average
actually doesn’t compare atall well with a defocussedlens– Most obvious difference is
that a single point of lightviewed in a defocussed lenslooks like a fuzzy blob; butthe averaging process wouldgive a little square.
• A Gaussian gives a goodmodel of a fuzzy blob
€
For a filter size N by M,Rij = HuvFi−u, j−v
v=1:M∑
u=1:N∑
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€
exp −x2 + y2
2σ2
An Isotropic Gaussian
• The picture shows asmoothing kernelproportional to
(which is a reasonable model ofa circularly symmetric fuzzyblob)
H ~
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Smoothing with a Gaussian
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Differentiation and convolution
• Recall
• Now this is linear andshift invariant, so mustbe the result of aconvolution.
• We could approximate thisas
• This is a convolution: ( butit’s not a very good way to dothings, as we shall see)
€
∂f∂x
= limε→0
f x + ε, y( )ε
−f x, y( )ε
€
∂f∂x
≈f xn+1,y( )− f xn , y( )
Δx
€
∂h∂x
≈ hi+1, j − hi−1, j ⇒∂h∂x
= Di−u, j−vv∑
u∑ Fu,v
D =
0 0 01 0 −10 0 0
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
Filtering as a Linear Transform
h = 1 4 6 4 1
F =
F convolved with H = ?
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
1 4 6 4 1 0 0 0 0 0 0 0 0 0 0 00 1 4 6 4 1 0 0 0 0 0 0 0 0 0 00 0 1 4 6 4 1 0 0 0 0 0 0 0 0 00 0 0 1 4 6 4 1 0 0 0 0 0 0 0 00 0 0 0 1 4 6 4 1 0 0 0 0 0 0 00 0 0 0 0 1 4 6 4 1 0 0 0 0 0 00 0 0 0 0 0 1 4 6 4 1 0 0 0 0 00 0 0 0 0 0 0 1 4 6 4 1 0 0 0 00 0 0 0 0 0 0 0 1 4 6 4 1 0 0 00 0 0 0 0 0 0 0 0 1 4 6 4 1 0 00 0 0 0 0 0 0 0 0 0 1 4 6 4 1 00 0 0 0 0 0 0 0 0 0 0 1 4 6 4 1
F’ = H*F
H =
H*F*H'
x & y filtering
y filtering
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
Image Synthesis with Fourier Basis
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005F here means the transformed function
F here means fourier transform
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
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Finite differences
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Noise• Simplest noise model
– independent stationaryadditive Gaussian noise
– the noise value at eachpixel is given by anindependent draw fromthe same normalprobability distribution
• Issues– this model allows noise values
that could be greater thanmaximum camera output or lessthan zero
– for small standard deviations, thisisn’t too much of a problem - it’s afairly good model
– independence may not be justified(e.g. damage to lens)
– may not be stationary (e.g.thermal gradients in the ccd)For an image F,
the measured value G:
€
Gu,v = Fu,v + nu,v
n→DN(µ,σ 2) =
1
2πσ 2e−
1
2σ 2x−µ( )2
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sigma=1
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sigma=16
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Finite differences and noise• Finite difference filters
respond strongly to noise– obvious reason: image noise
results in pixels that look verydifferent from theirneighbours
• Generally, the larger thenoise the stronger theresponse
• What is to be done?– intuitively, most pixels in
images look quite a lot liketheir neighbours
– this is true even at an edge;along the edge they’re similar,across the edge they’re not
– suggests that smoothing theimage should help, by forcingpixels different to theirneighbours (=noise pixels?)to look more like neighbours
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Finite differences responding to noise
Increasing noise ->(this is zero mean additive gaussian noise)
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The response of a linear filter to noise
• Do only stationary independent additive Gaussian noise withzero mean (non-zero mean is easily dealt with)
• Generalized Average (Mean):– output is a weighted sum of inputs
– so we want mean of a weighted sum of zero mean normal randomvariables
– must be zero
€
µ = wixii=1
N∑ wi =1
i=1
N∑
Ri, j = wi−u, j−vGu,vu,v∑ = wi−u, j−v Fu,v + nu,v( )u,v∑
Ri, j = wi−u, j−vFu,v + wi−u, j−vnu,vu,v∑u,v∑Ri, j = ˆ F u,v + µu,v
€
Gu,v = Fu,v + nu,vn→
DN(µ,σ 2)1
2πσ 2e−
1
2σ 2x−µ( )2
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CSCI 5561: Computer Vision, Prof. Paul Schrater, Spring 2005
Response linear filter to noise
€
Var wi−u, j−vnu,vu,v∑
= E wi−u, j−v2 nu,v
2[ ]u,v∑
= wi−u, j−v2 E nu,v
2[ ]u,v∑= wi−u, j−v
2 σ 2u,v∑
=σ 2 wi−u, j−v2
u,v∑
• Variance:– recall
• variance of a sum of random variablesis sum of their variances
• variance of constant times randomvariable is constant^2 times variance
– then if σ is noise variance andkernel is w, variance of response is
€
Properties of sums of Gaussian random variables1N
ni = µi=1:N∑
Var 1N
nii=1:N∑
=
1N
E ni2[ ] =
i=1:N∑ 1
Nσ i
2
i=1:N∑
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Filter responses are correlated
• over scales similar to the scale of the filter
• Filtered noise is sometimes useful– looks like some natural textures, can be used to simulate
fire, etc.
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Smoothing reduces noise• Generally expect pixels to
“be like” their neighbours– surfaces turn slowly
– relatively few reflectancechanges
• Generally expect noiseprocesses to beindependent from pixel topixel
• Implies that smoothingsuppresses noise, forappropriate noise models
• Scale– the parameter in the
symmetric Gaussian– as this parameter goes up,
more pixels are involved inthe average
– and the image gets moreblurred
– and noise is more effectivelysuppressed
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The effects of smoothing Each row shows smoothingwith gaussians of differentwidth; each column showsdifferent realisations of an image of gaussian noise.