3-D Computater Vision CSc 83020

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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 VisionCSc 83020
• Revisit filtering (Gaussian and Median)
• Introduction to edge detection

3-D Computer Vision CSc83020 / Ioannis Stamos

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
• 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
• 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

Original: Mandrill

Smoothed withGaussian kernel

3-D Computer Vision CSc83020 / Ioannis Stamos

Gaussian Filters
• One-dimensional Gaussian
• Two-dimensional Gaussian

3-D Computer Vision CSc83020 / Ioannis Stamos

Gaussian Filters

3-D Computer Vision CSc83020 / Ioannis Stamos

Gaussian Filters

3-D Computer Vision CSc83020 / Ioannis Stamos

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

3-D Computer Vision CSc83020 / Ioannis Stamos

Edges & Edge Detection
• What are Edges?
• Theory of Edge Detection.
• Edge Detection in the Brain?
• Edge Detection using Resolution Pyramids

3-D Computer Vision CSc83020 / Ioannis Stamos

Edges

3-D Computer Vision CSc83020 / Ioannis Stamos

What are Edges?

Rapid Changes of intensity in small region

3-D Computer Vision CSc83020 / Ioannis Stamos

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

3-D Computer Vision CSc83020 / Ioannis Stamos

Edge easy to find

What is an Edge?

3-D Computer Vision CSc83020 / Ioannis Stamos

What is an Edge?

Where is edge? Single pixel wide or multiple pixels?

3-D Computer Vision CSc83020 / Ioannis Stamos

What is an Edge?

Noise: have to distinguish noise from actual edge

3-D Computer Vision CSc83020 / Ioannis Stamos

What is an Edge?

Is this one edge or two?

3-D Computer Vision CSc83020 / Ioannis Stamos

What is an Edge?

Texture discontinuity

3-D Computer Vision CSc83020 / Ioannis Stamos

Local Edge Detection

3-D Computer Vision CSc83020 / Ioannis Stamos

Edge Types

Ideal Step Edges

Ideal Ridge Edges

Ideal Roof Edges

Real Edges

I

x

Problems: Noisy Images

Discrete Images

3-D Computer Vision CSc83020 / Ioannis Stamos

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
• Noise smoothing
• Edge Enhancement
• Edge Localization
• Nonmaximum suppression
• Thresholding

3-D Computer Vision CSc83020 / Ioannis Stamos

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 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 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 Detection

y

B1,L(x,y)>0

t

B2,L(x,y)<0

x

Edge Magnitude

Edge Orientation

Rotationally Symmetric, Non-Linear

3-D Computer Vision CSc83020 / Ioannis Stamos

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

Ii,j+1

Ii+1,j+1

ε

Ii,j

Ii+1,j

Finite Difference Approximations

3-D Computer Vision CSc83020 / Ioannis Stamos

y

x

3-D Computer Vision CSc83020 / Ioannis Stamos

[Roberts ’65]

if

threshold then we have an edge

3-D Computer Vision CSc83020 / Ioannis Stamos

Mean filter convolved with first derivative filter

3-D Computer Vision CSc83020 / Ioannis Stamos

Examples

First derivative

Sobel operator

3-D Computer Vision CSc83020 / Ioannis Stamos

Second Derivative

Edge occurs at the zero-crossing of the second derivative

3-D Computer Vision CSc83020 / Ioannis Stamos

Laplacian
• Rotationally symmetric
• Linear computation (convolution)

3-D Computer Vision CSc83020 / Ioannis Stamos

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 Laplacian

More accurate

• Rotationally symmetric
• Linear computation (convolution)

3-D Computer Vision CSc83020 / Ioannis Stamos

Discrete Laplacian

Laplacian of an image

3-D Computer Vision CSc83020 / Ioannis Stamos

Discrete Laplacian

Laplacian is sensitive to noise

First smooth image with Gaussian

3-D Computer Vision CSc83020 / Ioannis Stamos

From Forsyth & Ponce.

3-D Computer Vision CSc83020 / Ioannis Stamos

From

Shree

Nayar’s

notes.

3-D Computer Vision CSc83020 / Ioannis Stamos

Discrete Laplacian w/ Smoothing

3-D Computer Vision CSc83020 / Ioannis Stamos

From

Shree

Nayar’s

notes.

3-D Computer Vision CSc83020 / Ioannis Stamos

Difference Operators – Second Derivative

3-D Computer Vision CSc83020 / Ioannis Stamos

From Forsyth & Ponce.

3-D Computer Vision CSc83020 / Ioannis Stamos

Edge Detection – Human Vision

LoG convolution in the brain – biological evidence!

Flipped LoG

LoG

3-D Computer Vision CSc83020 / Ioannis Stamos

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.

From

Forsyth

& Ponce.

3-D Computer Vision CSc83020 / Ioannis Stamos