3-D Computer Vision CSc83020 / Ioannis Stamos Revisit filtering (Gaussian and Median) Introduction...

3-D Computer Vision CSc83020 / Ioannis Stamos3-D Computer Vision CSc83020 / Ioannis Stamos

Revisit filtering (Gaussian and Median)Revisit filtering (Gaussian and Median) Introduction to edge detectionIntroduction to edge detection

3-D Computater Vision3-D Computater VisionCSc 83020CSc 83020


Linear FiltersLinear Filters

Given an image Given an image InIn((xx,,yy) generate a) generate anew image new image OutOut((xx,,yy):): For each pixel (For each pixel (xx,,yy))OutOut((xx,,yy) is a linear combination of pixels) is a linear combination of pixelsin the neighborhood of in the neighborhood of InIn((xx,,yy))

This algorithm isThis algorithm is Linear in input intensityLinear in input intensity Shift invariantShift invariant


Discrete ConvolutionDiscrete Convolution

This is the discrete analogue of convolutionThis is the discrete analogue of convolution The pattern of weights is called the “kernel”The pattern of weights is called the “kernel”

of the filterof the filter Will be useful in smoothing, edge detectionWill be useful in smoothing, edge detection


Computing ConvolutionsComputing Convolutions

What happens near edges of image?What happens near edges of image? Ignore (Ignore (OutOut is smaller than is smaller than InIn)) Pad with zeros (edges get dark)Pad with zeros (edges get dark) Replicate edge pixelsReplicate edge pixels Wrap aroundWrap around ReflectReflect Change filterChange filter

i j

jyixInjifyxOut ),(),(),( i j

jyixInjifyxOut ),(),(),(


Example: SmoothingExample: Smoothing

Original: MandrillOriginal: Mandrill Smoothed withSmoothed withGaussian kernelGaussian kernel


Gaussian FiltersGaussian Filters

One-dimensional GaussianOne-dimensional Gaussian

Two-dimensional GaussianTwo-dimensional Gaussian

2

2

21

2

1)(

x

exG

2

2

21

2

1)(

x

exG

2

22

22

2

1),(

yx

eyxG

2

22

22

2

1),(

yx

eyxG



Gaussians are used because:Gaussians are used because: SmoothSmooth Decay to zero rapidlyDecay to zero rapidly Simple analytic formulaSimple analytic formula Limit of applying multiple filters is GaussianLimit of applying multiple filters is Gaussian

(Central limit theorem)(Central limit theorem) Separable:Separable:

G G22(x,y) = G(x,y) = G11(x) G(x) G11(y)(y)


Size of the maskSize of the mask


Edges & Edge DetectionEdges & Edge Detection

What are Edges?What are Edges? Theory of Edge Detection.Theory of Edge Detection. Edge Operators (Convolution Masks)Edge Operators (Convolution Masks) Edge Detection in the Brain?Edge Detection in the Brain? Edge Detection using Resolution PyramidsEdge Detection using Resolution Pyramids


EdgesEdges


What are Edges?What are Edges?

Rapid Changes of intensity in small region


What are Edges?What are Edges?

Surface-Normal discontinuity

Depth discontinuity

Surface-Reflectance Discontinuity

Illumination Discontinuity

Rapid Changes of intensity in small region


Local Edge DetectionLocal Edge Detection


What is an Edge?What is an Edge?

Edge easy to findEdge easy to find



Where is edge? Single pixel wide or multiple pixels?Where is edge? Single pixel wide or multiple pixels?



Noise: have to distinguish noise from actual edgeNoise: have to distinguish noise from actual edge



Is this one edge or two?Is this one edge or two?



Texture discontinuityTexture discontinuity


Local Edge DetectionLocal Edge Detection

Edge TypesEdge Types

Ideal Step Edges

Ideal Ridge Edges

Ideal Roof Edges


Real EdgesReal EdgesI

x

Problems: Noisy Images Discrete Images


Real EdgesReal EdgesWe want an Edge Operator that produces:

Edge Magnitude (strength)Edge directionEdge normalEdge position/center

High detection rate & good localization


The 3 steps of Edge DetectionThe 3 steps of Edge Detection

Noise smoothingNoise smoothing Edge EnhancementEdge Enhancement Edge LocalizationEdge Localization

Nonmaximum suppressionNonmaximum suppression ThresholdingThresholding


Theory of Edge DetectionTheory of Edge Detection

x

y B1,L(x,y)>0

B2,L(x,y)<0t

Unit Step Function:

t

dsstu

t

t

t

tu

)()(

0,0

0,2/1

0,1

)(



x

y B1,L(x,y)>0

B2,L(x,y)<0t

Unit Step Function:

t

dsstu

t

t

t

tu

)()(

0,0

0,2/1

0,1

)(

Ideal Edge: 0cossin),( yxyxL

Image Intensity (Brightness):

)cossin()12(1),( yxuBBByxI



x

y B1,L(x,y)>0

B2,L(x,y)<0t

Partial Derivatives:

)cossin()12(cos/

)cossin()12(sin/

yxBByI

yxBBxI

Directional!



x

y B1,L(x,y)>0

B2,L(x,y)<0t

222 )]cossin()12[()/()/(),( yxBByIxIyxS

Rotationally Symmetric, Non-Linear

),( yxSEdge Magnitude

)/,/tan( yIxIa Edge Orientation

Squared Gradient:


x

y B1,L(x,y)>0

B2,L(x,y)<0t

Laplacian:

)cossin()12(// '22222 yxBByIxII

(Rotationally Symmetric & Linear)

I

x

xI /

x

xI 22 /

Zero Crossing


Difference OperatorsDifference OperatorsIi,j+1 Ii+1,j+1

Ii,j Ii+1,jε

))())((2/1(/

))())((2/1(/

,1,,11,1

,,11,1,1

jijijiji

jijijiji

IIIIyI

IIIIxI

Finite Difference Approximations


Squared GradientSquared Gradient

x

y

),( yxf

22

y

f

x

ff


Squared GradientSquared Gradient

11

11

11

11

2

1

22

21if threshold then we have an edge

[Roberts ’65]


Squared GradientSquared Gradient– Sobel – Sobel

101

202

101

11

11

11

111

121

000

121

11

11

11

112

Mean filter convolved with first derivative filter


ExamplesExamples

First derivative Sobel operator


Second DerivativeSecond Derivative

f

f

f

Edge occurs at the zero-crossing of the second derivative


LaplacianLaplacian

2

2

2

22

y

f

x

f

• Rotationally symmetric• Linear computation (convolution)


Discrete LaplacianDiscrete Laplacian

Ii,j+1 Ii+1,j+1

Ii,j Ii+1,j

)2)(/1(/

)2)(/1(/

1,,1,222

,1,,1222

jijiji

jijiji

IIIyI

IIIxI

Finite Difference Approximations

Ii+1,j-1Ii,j-1Ii-1,j-1

Ii-1,j

Ii-1,j+1



• Rotationally symmetric• Linear computation (convolution)

010

141

0102

141

4204

141

6

12

More accurate



Laplacian of an image



Laplacian is sensitive to noiseFirst smooth image with Gaussian


From Forsyth & Ponce.


FromShreeNayar’snotes.


Discrete Laplacian w/ Discrete Laplacian w/ SmoothingSmoothing

IGIGI 22LoG


FromShreeNayar’snotes.


Difference Operators – Second Difference Operators – Second DerivativeDerivative

IGIGI 22LoG

00100

02210

121621

01210

00100


Edge Detection – Human VisionEdge Detection – Human Vision

LoG convolution in the brain – biological evidence!

LoGFlipped LoG

Image Resolution PyramidsImage Resolution Pyramids

Can save computations.Consolidation: Average pixels at one level to find

value at higher level.Template Matching: Find match in COARSE resolution.

Then move to FINER resolution.

3-D Computer Vision CSc83020 / Ioannis Stamos Revisit filtering (Gaussian and Median) Introduction...

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