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3-D Computater Vision CSc 83020. Revisit filtering (Gaussian and Median) Introduction to edge detection. Linear Filters. Given an image In ( x , y ) generate a new image Out ( x , y ):

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3 d computater vision csc 83020
3-D Computater VisionCSc 83020
  • Revisit filtering (Gaussian and Median)
  • Introduction to edge detection

3-D Computer Vision CSc83020 / Ioannis Stamos

linear filters
Linear Filters
  • Given an image In(x,y) generate anew image Out(x,y):
    • For each pixel (x,y)Out(x,y) is a linear combination of pixelsin the neighborhood of In(x,y)
  • This algorithm is
    • Linear in input intensity
    • Shift invariant

3-D Computer Vision CSc83020 / Ioannis Stamos

discrete convolution
Discrete Convolution
  • This is the discrete analogue of convolution
  • The pattern of weights is called the “kernel”of the filter
  • Will be useful in smoothing, edge detection

3-D Computer Vision CSc83020 / Ioannis Stamos

computing convolutions
Computing Convolutions
  • What happens near edges of image?
    • Ignore (Out is smaller than In)
    • Pad with zeros (edges get dark)
    • Replicate edge pixels
    • Wrap around
    • Reflect
    • Change filter

3-D Computer Vision CSc83020 / Ioannis Stamos

example smoothing
Example: Smoothing

Original: Mandrill

Smoothed withGaussian kernel

3-D Computer Vision CSc83020 / Ioannis Stamos

gaussian filters
Gaussian Filters
  • One-dimensional Gaussian
  • Two-dimensional Gaussian

3-D Computer Vision CSc83020 / Ioannis Stamos

gaussian filters1
Gaussian Filters

3-D Computer Vision CSc83020 / Ioannis Stamos

gaussian filters2
Gaussian Filters

3-D Computer Vision CSc83020 / Ioannis Stamos

gaussian filters3
Gaussian Filters
  • Gaussians are used because:
    • Smooth
    • Decay to zero rapidly
    • Simple analytic formula
    • Limit of applying multiple filters is Gaussian(Central limit theorem)
    • Separable: G2(x,y) = G1(x) G1(y)

3-D Computer Vision CSc83020 / Ioannis Stamos

size of the mask
Size of the mask

3-D Computer Vision CSc83020 / Ioannis Stamos

edges edge detection
Edges & Edge Detection
  • What are Edges?
  • Theory of Edge Detection.
  • Edge Operators (Convolution Masks)
  • Edge Detection in the Brain?
  • Edge Detection using Resolution Pyramids

3-D Computer Vision CSc83020 / Ioannis Stamos

edges
Edges

3-D Computer Vision CSc83020 / Ioannis Stamos

what are edges
What are Edges?

Rapid Changes of intensity in small region

3-D Computer Vision CSc83020 / Ioannis Stamos

what are edges1
What are Edges?

Surface-Normal discontinuity

Depth discontinuity

Surface-Reflectance Discontinuity

Illumination Discontinuity

Rapid Changes of intensity in small region

3-D Computer Vision CSc83020 / Ioannis Stamos

local edge detection
Local Edge Detection

3-D Computer Vision CSc83020 / Ioannis Stamos

what is an edge

Edge easy to find

What is an Edge?

3-D Computer Vision CSc83020 / Ioannis Stamos

what is an edge1
What is an Edge?

Where is edge? Single pixel wide or multiple pixels?

3-D Computer Vision CSc83020 / Ioannis Stamos

what is an edge2
What is an Edge?

Noise: have to distinguish noise from actual edge

3-D Computer Vision CSc83020 / Ioannis Stamos

what is an edge3
What is an Edge?

Is this one edge or two?

3-D Computer Vision CSc83020 / Ioannis Stamos

what is an edge4
What is an Edge?

Texture discontinuity

3-D Computer Vision CSc83020 / Ioannis Stamos

local edge detection1
Local Edge Detection

3-D Computer Vision CSc83020 / Ioannis Stamos

edge types
Edge Types

Ideal Step Edges

Ideal Ridge Edges

Ideal Roof Edges

real edges
Real Edges

I

x

Problems: Noisy Images

Discrete Images

3-D Computer Vision CSc83020 / Ioannis Stamos

real edges1
Real Edges

We want an Edge Operator that produces:

Edge Magnitude (strength)

Edge direction

Edge normal

Edge position/center

High detection rate & good localization

3-D Computer Vision CSc83020 / Ioannis Stamos

the 3 steps of edge detection
The 3 steps of Edge Detection
  • Noise smoothing
  • Edge Enhancement
  • Edge Localization
    • Nonmaximum suppression
    • Thresholding

3-D Computer Vision CSc83020 / Ioannis Stamos

theory of edge detection
Theory of Edge Detection

Unit Step Function:

y

B1,L(x,y)>0

t

B2,L(x,y)<0

x

3-D Computer Vision CSc83020 / Ioannis Stamos

theory of edge detection1
Theory of Edge Detection

Unit Step Function:

y

B1,L(x,y)>0

t

B2,L(x,y)<0

x

Ideal Edge:

Image Intensity (Brightness):

3-D Computer Vision CSc83020 / Ioannis Stamos

theory of edge detection2
Theory of Edge Detection

Partial Derivatives:

y

B1,L(x,y)>0

t

B2,L(x,y)<0

Directional!

x

3-D Computer Vision CSc83020 / Ioannis Stamos

theory of edge detection3
Theory of Edge Detection

y

B1,L(x,y)>0

t

B2,L(x,y)<0

x

Squared Gradient:

Edge Magnitude

Edge Orientation

Rotationally Symmetric, Non-Linear

3-D Computer Vision CSc83020 / Ioannis Stamos

theory of edge detection4
Theory of Edge Detection

Laplacian:

y

B1,L(x,y)>0

t

B2,L(x,y)<0

x

(Rotationally Symmetric & Linear)

I

x

x

Zero Crossing

difference operators
Difference Operators

Ii,j+1

Ii+1,j+1

ε

Ii,j

Ii+1,j

Finite Difference Approximations

3-D Computer Vision CSc83020 / Ioannis Stamos

squared gradient
Squared Gradient

y

x

3-D Computer Vision CSc83020 / Ioannis Stamos

squared gradient1
Squared Gradient

[Roberts ’65]

if

threshold then we have an edge

3-D Computer Vision CSc83020 / Ioannis Stamos

squared gradient sobel
Squared Gradient– Sobel

Mean filter convolved with first derivative filter

3-D Computer Vision CSc83020 / Ioannis Stamos

examples
Examples

First derivative

Sobel operator

3-D Computer Vision CSc83020 / Ioannis Stamos

second derivative
Second Derivative

Edge occurs at the zero-crossing of the second derivative

3-D Computer Vision CSc83020 / Ioannis Stamos

laplacian
Laplacian
  • Rotationally symmetric
  • Linear computation (convolution)

3-D Computer Vision CSc83020 / Ioannis Stamos

discrete laplacian
Discrete Laplacian

Ii,j+1

Ii+1,j+1

Ii-1,j+1

Ii,j

Ii+1,j

Ii-1,j

Ii-1,j-1

Ii,j-1

Ii+1,j-1

Finite Difference Approximations

3-D Computer Vision CSc83020 / Ioannis Stamos

discrete laplacian1
Discrete Laplacian

More accurate

  • Rotationally symmetric
  • Linear computation (convolution)

3-D Computer Vision CSc83020 / Ioannis Stamos

discrete laplacian2
Discrete Laplacian

Laplacian of an image

3-D Computer Vision CSc83020 / Ioannis Stamos

discrete laplacian3
Discrete Laplacian

Laplacian is sensitive to noise

First smooth image with Gaussian

3-D Computer Vision CSc83020 / Ioannis Stamos

slide42

From Forsyth & Ponce.

3-D Computer Vision CSc83020 / Ioannis Stamos

slide43

From

Shree

Nayar’s

notes.

3-D Computer Vision CSc83020 / Ioannis Stamos

discrete laplacian w smoothing
Discrete Laplacian w/ Smoothing

3-D Computer Vision CSc83020 / Ioannis Stamos

slide45

From

Shree

Nayar’s

notes.

3-D Computer Vision CSc83020 / Ioannis Stamos

difference operators second derivative
Difference Operators – Second Derivative

3-D Computer Vision CSc83020 / Ioannis Stamos

slide47

From Forsyth & Ponce.

3-D Computer Vision CSc83020 / Ioannis Stamos

edge detection human vision
Edge Detection – Human Vision

LoG convolution in the brain – biological evidence!

Flipped LoG

LoG

3-D Computer Vision CSc83020 / Ioannis Stamos

image resolution pyramids
Image 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.

slide50

From

Forsyth

& Ponce.

3-D Computer Vision CSc83020 / Ioannis Stamos