EECS 274 Computer Vision Linear Filters and Edges.

EECS 274 Computer Vision

Linear Filters and Edges

Linear filters and edges

• Linear filters• Scale space• Gaussian pyramid and wavelets• Edges

• Reading: Chapters 7 and 8 of FP, Chapter 3 of S

Linear filters

• General process:– Form new image whose

pixels are a weighted sum of original pixel values, using the same set of weights at each point.

• Properties– Output is a linear function

of the input– Output is a shift-invariant

function of the input (i.e. shift the input image two pixels to the left, the output is shifted two pixels to the left)

• Example: smoothing by averaging– form the average of pixels

in a neighbourhood

• Example: smoothing with a Gaussian– form a weighted average

of pixels in a neighbourhood

• Example: finding a derivative– form a weighted average

of pixels in a neighbourhood

vuuv

kjv

kjvuv

kiu

kiuij F

kF

kR

,22 )12(

1

)12(

1

Convolution

• Represent these weights as an image, H

• H is usually called the kernel

• Operation is called convolution– it’s associative

• Notation in textbook:

• Notice wierd order of indices– all examples can be put in this

form

– it’s a result of the derivation expressing any shift-invariant linear operator as a convolution.

FHR

FHRvu

vuvjuiij

,

,,

hfg

hfg

lkhljkifjig

lkhljkifjig

lk

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*

),(),(),(

),(),(),(

,

,

Example: smoothing by averaging

Smoothing with a Gaussian

• Smoothing with an average actually doesn’t compare at all well with a defocussed lens– Most obvious difference is

that a single point of light viewed in a defocussed lens looks like a fuzzy blob; but the averaging process would give a little square.

• A Gaussian gives a good model of a fuzzy blob

An isotropic Gaussian• The picture shows a

smoothing kernel proportional to

(which is a reasonable model of a circularly symmetric fuzzy blob)

2

22

2 2exp

2

1),(

yx

yxG

Smoothing with a Gaussian

Differentiation and convolution• Recall

• Now this is linear and shift invariant, so must be the result of a convolution

• We could approximate this as

(which is obviously a convolution; it’s not a very good way to do things, as we shall see)

),(),(lim

0

yxfyxf

x

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x

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x

f nn

),(),( 1

000

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000

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Finite differences

Partial derivative in y axis, respond strongly tohorizontal edges

Partial derivative in x axis, respond strongly to vertical edges

Spatial filter

• Approximation

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|)||)(|||

])()[(||

||,

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Roberts operator

One of the earliest edge detection algorithm by Lawrence Roberts

Sobel operatorOne of the earliest edge detection algorithm by Irwine Sobel

Noise

• Simplest noise model– independent stationary

additive Gaussian noise– the noise value at each

pixel is given by an independent draw from the same normal probability distribution

• Issues– this model allows noise

values that could be greater than maximum camera output or less than zero

– for small standard deviations, this isn’t too much of a problem - it’s a fairly good model

– independence may not be justified (e.g. damage to lens)

– may not be stationary (e.g. thermal gradients in the ccd)

sigma=1

sigma=16

Finite differences and noise

• Finite difference filters respond strongly to noise– obvious reason: image

noise results in pixels that look very different from their neighbours

• Generally, the larger the noise the stronger the response

• What is to be done?– intuitively, most pixels in

images look quite a lot like their neighbours

– this is true even at an edge; along the edge they’re similar, across the edge they’re not

– suggests that smoothing the image should help, by forcing pixels different to their neighbours (=noise pixels?) to look more like neighbours

Finite differences responding to noise

Increasing noise -> (this is zero mean additive Gaussian noise)Difference operation is strongly influenced by noise (the image is increasingly grainy)

σ=0.03 σ=0.09

The response of a linear filter to noise• Do only stationary

independent additive Gaussian noise with zero mean (non-zero mean is easily dealt with)

• Mean:– output is a weighted sum

of inputs– so we want mean of a

weighted sum of zero mean normal random variables

– must be zero

• Variance:– recall

• variance of a sum of random variables is sum of their variances

• variance of constant times random variable is constant^2 times variance

– then if is noise variance and kernel is K, variance of response is

vu

vuK

,

22

,

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.

Smoothed noise

Smoothing stationary additive Gaussian noise results in signals where pixel values tend to increasingly similar to the value of neighboring pixels (as filter kernel causes correlation)

Smoothing reduces noise

• Generally expect pixels to “be like” their neighbours– surfaces turn slowly– relatively few reflectance

changes

• Generally expect noise processes to be independent from pixel to pixel

• Implies that smoothing suppresses noise, for appropriate noise models

• Scale– the parameter in the

symmetric Gaussian– as this parameter goes up,

more pixels are involved in the average

– and the image gets more blurred

– and noise is more effectively suppressed

The effects of smoothing Each row shows smoothingwith gaussians of differentwidth; each column showsdifferent realisations of an image of gaussian noise.

Gradients and edges

• Points of sharp change in an image are interesting:– change in reflectance– change in object– change in illumination– noise

• Sometimes called edge points

• General strategy– determine image gradient

– now mark points where gradient magnitude is particularly large wrt neighbours (ideally, curves of such points).

In one dimension, the 2nd derivative of a signal is zero when thederivative magnitude is extremal a good place to look for edgeis where the second derivative is zero.

Smoothing and differentiation• Issue: noise

– smooth before differentiation– two convolutions to smooth, then differentiate?– actually, no - we can use a derivative of

Gaussian filter• because differentiation is convolution, and

convolution is associative

The scale of the smoothing filter affects derivative estimates, and alsothe semantics of the edges recovered.

1 pixel 3 pixels 7 pixels

Gradient• Gradient equation:

• Represents direction of most rapid change in intensity

• Gradient direction:

• The edge strength is given by the gradient magnitude

Theory of Edge Detection

Ideal edge

Unit step function:

Image intensity (brightness):

dsstut

0for 0

0for 21

0for 1

t

t

t

tu

0,:

0,:

0cossin,

2

1

yxLB

yxLB

yxyxL

cossin, 121 yxuBBByxI

• Image intensity (brightness):

• Partial derivatives (gradients):

• Squared gradient:

Edge Magnitude:

Edge Orientation:

Rotationally symmetric, non-linear operator

(normal of the edge)



cossincos

cossinsin

12

12

yxBBy

I

yxBBx

I

212

22

cossin,

yxBBy

I

x

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yxs ,

x

I

y

I/arctan

• Image intensity (brightness):

• Partial derivatives (gradients):

• Laplacian:

Rotationally symmetric, linear operator

zero-crossing



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12

12

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I

yxBBx

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cossin'122

2

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22 yxBB

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Discrete Edge Operators

• How can we differentiate a discrete image?

Finite difference approximations:

1, jiI 1,1 jiI

jiI , jiI ,1

Convolution masks :

jijijiji

jijijiji

IIIIy

I

IIIIx

I

,1,,11,1

,,11,1,1

2

12

1

21

x

I

21

y

I

1, jiI 1,1 jiI

jiI , jiI ,1

1,1 jiI

jiI ,1

1,1 jiI 1, jiI 1,1 jiI

• Second order partial derivatives:

• Laplacian :

Convolution masks :

or

Discrete Edge Operators

(more accurate)

1 4 1

4 -20

4

1 4 1

0 1 0

1 -4 1

0 1 0

1,,1,22

2

,1,,122

2

21

21

jijiji

jijiji

IIIy

I

IIIx

I

2

2

2

22

y

I

x

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22 1

I

26

1

Effects of Noise• Consider a single row or column of the image

– Plotting intensity as a function of position gives a signal

Where is the edge??

Where is the edge?

Solution: Smooth First

Look for peaks in

Derivative Theorem of Convolution…saves us one operation.

Laplacian of Gaussian (LoG)

Laplacian of Gaussian operator

Where is the edge? Zero-crossings of bottom graph !

Laplacian of Gaussian fh

xfh

x

2

2

2

2

2D Gaussian Edge Operators

Laplacian of GaussianGaussian Derivative of Gaussian (DoG)

Mexican Hat (Sombrero)• is the Laplacian operator:

sigma=2

sigma=4

contrast=1 contrast=4LOG zero crossingsscale

threshold

We still have unfortunate behaviorat corners and trihedral areas

There are three major issues: 1) The gradient magnitude at different scales is different; which should we choose? 2) The gradient magnitude is large along thick trail; how do we identify the significant points? 3) How do we link the relevant points up into curves?

σ=1 pixel σ=2 pixel

We wish to mark points along the curve where the magnitude is biggest.We can do this by looking for a maximum along a slice normal to the curve(non-maximum suppression). These points should form a curve. There arethen two algorithmic issues: at which point is the maximum, and where is thenext one?

Non-maximum Suppression

• Check if pixel is local maximum along gradient direction– requires checking interpolated pixels p and r

Predictingthe nextedge point

Assume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s).

Edge following: edge points occur along curve like chains

fine scale, high threshold σ=1 pixel

coarse scale, high threshold σ=4 pixels coarse scale, low threshold σ=4 pixels

Canny Edge Operator

• Smooth image I with 2D Gaussian:

• Find local edge normal directions for each pixel

• Compute edge magnitudes

• Locate edges by finding zero-crossings along the edge normal directions (non-maximum suppression)

IG

IG

IG

n

IG

0

2

2

n

IG

Original image magnitude of gradient

Canny edge detector

After non-maximum suppression

Canny edge detector

Canny with Canny with original

• The choice of depends on desired behavior– large detects large scale edges

– small detects fine features

Canny edge detector

Difference of Gaussians (DoG)• Laplacian of Gaussian can be approximated by

the difference between two different Gaussians

DoG Edge Detection

(a) (b) (b)-(a)

Unsharp Masking

200 400 600 800

100

200

300

400

500

– =

=+ a

blurred positive increase details in bright area

Edge Thresholding• Standard Thresholding:

• Can only select “strong” edges.

• Does not guarantee “continuity”.

• Hysteresis based Thresholding (use two thresholds)

Example: For “maybe” edges, decide on the edge if neighboring pixel is a strong edge.

Remaining issues

• Check that maximum value of gradient value is sufficiently large– drop-outs? use hysteresis

• use a high threshold to start edge curves and a low threshold to continue them.

Notice

• Something nasty is happening at corners

• Scale affects contrast• Edges aren’t bounding contours

Scale space

• Framework for multi-scale signal processing

• Define image structure in terms of scale with kernel size, σ

• Find scale invariant operations• See Scale-theory in computer vision by

Tony Lindeberg

Orientation representations

• The gradient magnitude is affected by illumination changes– but it’s direction isn’t

• We can describe image patches by the swing of the gradient orientation

• Important types:– constant window

• small gradient mags– edge window

• few large gradient mags in one direction

– flow window• many large gradient

mags in one direction– corner window

• large gradient mags that swing

Representing windows

• Types– constant

• small eigenvalues– edge

• one medium, one small– flow

• one large, one small– corner

• two large eigenvalues

window

T

window

Iy

GI

y

GI

y

GI

x

G

Iy

GI

x

GI

x

GI

x

G

IIH

))((

Looking at variations

Plotting in ellipses to understand the matrix (variation of gradient) of 3 × 3 window

Major and minor axes are alongthe eigenvectors of H, and the extentcorresponds to the size of eigenvalues

),(),( 1 yxHyx T

Plotting in ellipses to understand the matrix (variation of gradient) of 5 × 5 window

Corners

• Harris corner detector• Moravec corner detector• SIFT descriptors

Filters are templates

• Applying a filter at some point can be seen as taking a dot-product between the image and some vector

• Filtering the image is a set of dot products

• Insight – filters look like the effects

they are intended to find– filters find effects they

look like

convolution is equivalent totaking the dot product of the filterwith an image patch

derivative of Gaussian used as edgedetection

vuvuvuij

vuvuvjuiij

FHR

FHR

,,,

,,,

Normalized correlation

• Think of filters of a dot product– now measure the angle– i.e normalised correlation

output is filter output, divided by root sum of squares of values over which filter lies

– cheap and efficient method for finding patterns

• Tricks:– ensure that filter has a

zero response to a constant region (helps reduce response to irrelevant background)

– subtract image average when computing the normalizing constant (i.e. subtract the image mean in the neighbourhood)

– absolute value deals with contrast reversal

Positive responses

Zero mean image, -1:1 scale Zero mean image, -max:max scale

Anistropic scaling

• Symmetric Gaussian smoothing tends ot blur out edges rather aggressively

• Prefer an oriented smoothing operator that smoothes– aggressively perpendicular to the gradient– little along the gradient

• Also known as edge preserving smoothing• Formulated with diffusion equation

Diffusion equation for anistropic filter

• PDE that describes fluctuations in a material undergoing diffusion

• For istropic filter

• For anistropic filter

),(),(),(

trrDt

tr

condition initial as ),()0,,(

cases istropicfor ))),,(( 22

2

2

22

yxIyx

yxCyxc

smoothing no ,0),,( If

before asjust ,1),,( If

)),,((),,())),,(( 2

yxc

yxc

yxcyxcyxc

Anistropic filtering

Edge preserving filter

• Bilateral filter– Replace pixel’s value by a weighted

average of its neighbors in both space and intensity

One iteration Multiple iterations

EECS 274 Computer Vision Linear Filters and Edges.

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